statistics and probability essay example

Reflection about Statistics and Probability – Essay Sample [New]

Reflection paper about statistics and probability: introduction, what i learned in statistics and probability, reflection about statistics and probability: data analysis, reflection paper about statistics: future studies, statistics reflection paper: conclusion.

Learning statistics is viewed as an essential subject. It is also crucial to do statistics and probability reflection about their role in math, data management, and one’s daily life as a student. This statistics essay sample is going to cover what I have learned in statistics and probability. Essay will talk about my experience of learning the necessary skills for analyzing data and predicting possible outcomes.

I have extensively studied statistics and probability throughout this course. The probability course appeared to be a useful tool to apply in areas of statistical analysis. However, the part of learning statistics is much more prominent.

Statistical knowledge for both statisticians and non-statisticians is essential (Broers, 2006). So it is recommended that people from all fields be given the necessary statistical skills.

For that kind of reason, to gain quantitative skills to be applied and worked on in many ways, I followed this course. In this regard, I had hoped to acquire knowledge in designing experiments. I had wanted to grow in collecting and analyzing data, interpreting results, and drawing conclusions as well (Broers, 2006). I summarized and analyzed the results in my reflection about statistics and probability.

I can proudly say now that I have learned many useful things. I now understand math applications very clearly. I know how to collect, arrange, and explain the details. In data analysis, I may apply central tendency measurements such as mean, mode, and median.

I may also use dispersion measures to explain the data, such as standard deviation and variance. Furthermore, I have a detailed understanding of probability distribution. I can see what conditions are to be fulfilled for a normal distribution (Broers, 2006).

For example, I am aware of conditional probability and its applications. I also know how to use The Poisson process, Brownian motion process, Stochastic processes, Stationary processes, and Markovian processes. I learned the Ehrenfest model of diffusion, the symmetric random walk, queuing models, insurance risk theory, and Martingale theory.

Now, I can determine the relation between the two data sets. I can distinguish dependent and independent variables as well as the sort of relationship between them. I will tell you whether one random variable is causal to another. I am also able to determine positive, negative, and minimal correlations (Broers, 2006).

In most instances, data collection on an entire population is delicate (Chance, 2002). I got the necessary skills in sampling techniques in this respect. I have the skills to analyze sample data. Now I can draw inferences about the entire population using statistical probabilities and hypothesis testing.

In hypothesis testing, one seeks to determine whether the outcomes of a given sample are due to chance or known cause (Chance, 2002). Knowledge is used in the implementation of significance level, critical value, degrees of freedom, and p-value. One must be able to present the null hypothesis and the alternative hypotheses (Chance, 2002).

Now I’m able to use a t-test to assess if there are statistically significant variations between two data sets. In this regard, I understand the required assumptions for the t-test to be applied. I have a clear insight into the analysis of variance (ANOVA), both single and bidirectional. However, I feel that further practice would improve my knowledge of all of those applications (Chance, 2002).

This program has provided me with a good understanding of how statistics play a major part in life. Among other areas, statistics is the most commonly used research method in medicine, education, psychology, business, and economics (Rumsey, 2002). It helps to shape the choices people make in their everyday lives. Statistical studies may provide a clear picture of the consequences. For example, the results of such activities as smoking and contribute to corrective steps.

I was always keen to build a stable scientific career. I have now decided to major in statistics after taking this course. I would like to have advanced statistical skills that will help me to manage and evaluate complex research problems. The knowledge I’ve already obtained in this case will give me a strong foothold.

The goals I had hoped to accomplish by following this cause were well achieved. Now I can conduct experiments and collect, analyze, and interpret data. In real-life scenarios, I can apply that information and draw conclusions that will help develop answers to some issues.

Also, I have thoroughly studied and understood various causes of probability. I will be able to apply this knowledge where needed. Nevertheless, statistics will remain my main field of research.

This course, therefore, gave me the desire to seek additional statistical knowledge. For this reason, I intend to be in a better position in statistics to deal with more complex research issues.

• Broers, N. J. Learning goals: the primacy of statistical knowledge. 2006, Maastricht: Maastricht University.
• Chance, B. L. Components of statistical thinking and implications for instruction and assessment . 2002, Journal of Statistics Education, 10(3)
• Rumsey, D. J. Statistical literacy as a goal for introductory statistics courses. 2002, Journal of Statistics Education, 10(3)

What should I include in my statistics reflection paper?

Your statistics essay should contain a number of research data. Thoroughly research your subject. You can also include visual support like graphs or diagrams. Make sure the statistics are broken down and provide a general image of the issue.

Why are statistics so difficult?

Much of statistics makes no sense to students as it’s taught out of context. Many people often do not understand anything until they begin to examine data in their studies. You need to gain academic knowledge before you can understand statistics.

How do you start a reflection on statistics?

The good idea is to start your essay on statistics with a choice of the right topic. Make sure to research everything thoroughly and take notes of interesting observations. You should have a detailed understanding of a problem, and work well with data.

What is the aim of statistics?

Statistics aims to help you to use the right methods to gather the data. It ensures you use analysis correctly and present the results effectively. Statistics are essential to make science-based discoveries, make data-based decisions, and predict possible results.

What is the importance of statistics and probability?

Statistics is the mathematics that we use to gather, organize, and interpret numerical information. Probability is the study of possible events. It is often used in the analysis of chance games, genetics, and weather forecasting. A myriad of other everyday occurrences can be examined.

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Unit 7: Probability

About this unit.

Probability tells us how often some event will happen after many repeated trials. You've experienced probability when you've flipped a coin, rolled some dice, or looked at a weather forecast. Go deeper with your understanding of probability as you learn about theoretical, experimental, and compound probability, and investigate permutations, combinations, and more!

Basic theoretical probability

• Intro to theoretical probability (Opens a modal)
• Probability: the basics (Opens a modal)
• Simple probability: yellow marble (Opens a modal)
• Simple probability: non-blue marble (Opens a modal)
• Intuitive sense of probabilities (Opens a modal)
• The Monty Hall problem (Opens a modal)
• Simple probability Get 5 of 7 questions to level up!
• Comparing probabilities Get 5 of 7 questions to level up!

Probability using sample spaces

• Probability with counting outcomes (Opens a modal)
• Example: All the ways you can flip a coin (Opens a modal)
• Die rolling probability (Opens a modal)
• Subsets of sample spaces (Opens a modal)
• Subsets of sample spaces Get 3 of 4 questions to level up!

Basic set operations

• Intersection and union of sets (Opens a modal)
• Relative complement or difference between sets (Opens a modal)
• Universal set and absolute complement (Opens a modal)
• Subset, strict subset, and superset (Opens a modal)
• Bringing the set operations together (Opens a modal)
• Basic set notation Get 5 of 7 questions to level up!

Experimental probability

• Experimental probability (Opens a modal)
• Theoretical and experimental probabilities (Opens a modal)
• Making predictions with probability (Opens a modal)
• Simulation and randomness: Random digit tables (Opens a modal)
• Experimental probability Get 5 of 7 questions to level up!
• Making predictions with probability Get 5 of 7 questions to level up!

Randomness, probability, and simulation

• Experimental versus theoretical probability simulation (Opens a modal)
• Theoretical and experimental probability: Coin flips and die rolls (Opens a modal)
• Random number list to run experiment (Opens a modal)
• Random numbers for experimental probability (Opens a modal)
• Statistical significance of experiment (Opens a modal)
• Interpret results of simulations Get 3 of 4 questions to level up!

Addition rule

• Probability with Venn diagrams (Opens a modal)
• Addition rule for probability (Opens a modal)
• Addition rule for probability (basic) (Opens a modal)
• Adding probabilities Get 3 of 4 questions to level up!
• Two-way tables, Venn diagrams, and probability Get 3 of 4 questions to level up!

Multiplication rule for independent events

• Sample spaces for compound events (Opens a modal)
• Compound probability of independent events (Opens a modal)
• Probability of a compound event (Opens a modal)
• "At least one" probability with coin flipping (Opens a modal)
• Free-throw probability (Opens a modal)
• Three-pointer vs free-throw probability (Opens a modal)
• Probability without equally likely events (Opens a modal)
• Independent events example: test taking (Opens a modal)
• Die rolling probability with independent events (Opens a modal)
• Probabilities involving "at least one" success (Opens a modal)
• Sample spaces for compound events Get 3 of 4 questions to level up!
• Independent probability Get 3 of 4 questions to level up!
• Probabilities of compound events Get 3 of 4 questions to level up!
• Probability of "at least one" success Get 3 of 4 questions to level up!

Multiplication rule for dependent events

• Dependent probability introduction (Opens a modal)
• Dependent probability: coins (Opens a modal)
• Dependent probability example (Opens a modal)
• Independent & dependent probability (Opens a modal)
• The general multiplication rule (Opens a modal)
• Dependent probability (Opens a modal)
• Dependent probability Get 3 of 4 questions to level up!

Conditional probability and independence

• Calculating conditional probability (Opens a modal)
• Conditional probability explained visually (Opens a modal)
• Conditional probability using two-way tables (Opens a modal)
• Conditional probability tree diagram example (Opens a modal)
• Tree diagrams and conditional probability (Opens a modal)
• Conditional probability and independence (Opens a modal)
• Analyzing event probability for independence (Opens a modal)
• Calculate conditional probability Get 3 of 4 questions to level up!
• Dependent and independent events Get 3 of 4 questions to level up!

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• Essay Database >
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Sample Essay On Use Of Statistics And Probability In The Real World

Type of paper: Essay

Topic: Business , Gender , Decision , Information , Workplace , Study , Job , Probability

Published: 03/05/2020

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This paper contains an email addressed to the chief officer executive officer of American Intellectual Union (AIU). The email briefs the CEO about the distribution and probability of few selected variables based on of AIU’s Job Satisfaction Survey Data

The purpose of this email is to provide you with information concerning the distribution and probabilities selected variables based on the AIU job satisfaction survey data. The email also contains a brief discussion of the use of probability and statistics in real world and organizations.

Overview of the Data Set

The job satisfaction survey data included nine variables which can be categorized as either qualitative or quantitative. The qualitative variables were gender, department and position. On the other hand, the quantitative variables were age, tenure, intrinsic job satisfaction, extrinsic job satisfaction, overall job satisfaction and benefits.

Probability and statistics are used for decision making in real world. Statistical techniques and concepts are used collect, analyze and interpret data. This makes it possible for a researcher to establish trends, patterns and relationships among variables (Johnson & McAllister, 2009). The analyzed data provide evidence for supporting a position which is critical in decision making process. For instance analyzed data from census are used by states to make and support policies and interventions. Probability on the other hand is used to test assumptions and make inferences (Mendenhall, Beaver, & Beaver, 2013). For instance, it used clinical research to test hypothesis concerning the effectiveness of drugs.

The Value of Statistics

Statistics provide organizations with accurate, reliable and helpful information on its operations. This information is used for making decision on whether to employ additional staff, the quantity of stationery to purchase, and when to launch a new product among other decisions that organizations must make. Besides, the statistics is used for forecasting which help in planning (Holden, Peel & Thompson, 1990). Statistics is also used in quality control which enables organizations to maintain standards and minimize losses

Distributions

The survey sampled equal number of females and males to take part in the survey. In total there were 106 participants of which males and females were 53 each. This information has been captured in the table below in terms of percentages. The analysis of distribution of individuals by gender is important because gives information about the sampling procedures used and the reliability of the data.

Distribution of Individuals by Gender

Tenure with Company Distribution by Gender The following table shows the distribution of individuals by tenure and gender. It is based simple count. There were differences in number of males and females participants in every age group. The highest number of individuals had served for less than two years.

Percentage of the Survey Participants in Each Department

The following table shows the distribution of individuals by department given in terms of percentages. Sample Mean for Extrinsic Value by Gender The following table shows the extrinsic job satisfaction means for males and females. The mean of males was more than the mean of female by 0.1.

Probabilities

This section contains calculation of the probabilities for few variables of interest. These include the probability that an individual will be between 16–21 years of age., the probability that an individual’s overall job satisfaction is 5.2 or lower, the probability that an individual will be a female in the human resources department and the probability that an individual will be a salaried employee whose intrinsic satisfaction value.

Probabilities in the Business World

Probability is largely used in decision making in business world. This is done by getting the number of times an event has occurred and looking at the opportunities that comes with the occurrence of that event. For instance, the probability of failure of business startup can be used to make a decision on whether or not to invest in a particular sector. Besides, business uses probability distributions to estimate sales, revenue and profitability. For instance, a probability distribution for sales may be used by entrepreneur for scenario analysis. The worst scenario case is calculated using values from tail end of the probability distribution while a value in the middle of the distribution is used to compute worst best scenario case. Moreover, probability is used in risk assessment in business. This is done by looking at the loss and return probabilities for investments.

The information provided above is important for decision with regard to actions that must be taken to boost overall, intrinsic and extrinsic job satisfaction. It is also reveals data characteristics that are helpful in making conclusions. For instance, men are more extrinsically satisfied with their jobs than women. Sincerely, Business Administrator

Holden, K., Peel, D., & Thompson, J. L. (1990). Economic forecasting: An introduction. Cambridge [England: Cambridge University Press. Johnston, W., & McAllister, A. M. (2009). A transition to advanced mathematics: A survey course. Oxford: Oxford University Press. Mendenhall, W., Beaver, R. J., & Beaver, B. M. (2013). Introduction to probability and statistics. Pacific Grove, Calif: Brooks/Cole.

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7.6: Basic Concepts of Probability

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Learning Objectives

After completing this section, you should be able to:

• Define probability including impossible and certain events.
• Calculate basic theoretical probabilities.
• Calculate basic empirical probabilities.
• Distinguish among theoretical, empirical, and subjective probability.
• Calculate the probability of the complement of an event.

It all comes down to this. The game of Monopoly that started hours ago is in the home stretch. Your sister has the dice, and if she rolls a 4, 5, or 7 she’ll land on one of your best spaces and the game will be over. How likely is it that the game will end on the next turn? Is it more likely than not? How can we measure that likelihood? This section addresses this question by introducing a way to measure uncertainty.

Introducing Probability

Uncertainty is, almost by definition, a nebulous concept. In order to put enough constraints on it that we can mathematically study it, we will focus on uncertainty strictly in the context of experiments. Recall that experiments are processes whose outcomes are unknown; the sample space for the experiment is the collection of all those possible outcomes. When we want to talk about the likelihood of particular outcomes, we sometimes group outcomes together; for example, in the Monopoly example at the beginning of this section, we were interested in the roll of 2 dice that might fall as a 4, 5, or 7. A grouping of outcomes that we’re interested in is called an event . In other words, an event is a subset of the sample space of an experiment; it often consists of the outcomes of interest to the experimenter.

Once we have defined the event that interests us, we can try to assess the likelihood of that event. We do that by assigning a number to each event ( E E ) called the probability of that event ( P ( E ) P ( E ) ). The probability of an event is a number between 0 and 1 (inclusive). If the probability of an event is 0, then the event is impossible. On the other hand, an event with probability 1 is certain to occur. In general, the higher the probability of an event, the more likely it is that the event will occur.

Example 7.16

Determining certain and impossible events.

Consider an experiment that consists of rolling a single standard 6-sided die (with faces numbered 1-6). Decide if these probabilities are equal to zero, equal to one, or somewhere in between.

• P ( roll a 4 ) P ( roll a 4 )
• P ( roll a 7 ) P ( roll a 7 )
• P ( roll a positive number ) P ( roll a positive number )
• P ( roll a 1 3 ) P ( roll a 1 3 )
• P ( roll an even number ) P ( roll an even number )
• P ( roll a single-digit number ) P ( roll a single-digit number )

Let's start by identifying the sample space. For one roll of this die, the possible outcomes are {1, 2, 3, 4, 5,6}. We can use that to assess these probabilities:

• We see that 4 is in the sample space, so it’s possible that it will be the outcome. It’s not certain to be the outcome, though. So, 0 < P ( roll a 4 ) < 1 0 < P ( roll a 4 ) < 1 .
• Notice that 7 is not in the sample space. So, P ( roll a 7 ) = 0 P ( roll a 7 ) = 0 .
• Every outcome in the sample space is a positive number, so this event is certain. Thus, P ( roll a positive number ) = 1 P ( roll a positive number ) = 1 .
• Since 1 3 1 3 is not in the sample space, P ( roll a 1 3 ) = 0 P ( roll a 1 3 ) = 0 .
• Some outcomes in the sample space are even numbers (2, 4, and 6), but the others aren’t. So, 0 < P ( roll an even number ) < 1 0 < P ( roll an even number ) < 1 .
• Every outcome in the sample space is a single-digit number, so P ( roll a single-digit number ) = 1 P ( roll a single-digit number ) = 1 .

Your Turn 7.16

Three ways to assign probabilities.

The probabilities of events that are certain or impossible are easy to assign; they’re just 1 or 0, respectively. What do we do about those in-between cases, for events that might or might not occur? There are three methods to assign probabilities that we can choose from. We’ll discuss them here, in order of reliability.

Method 1: Theoretical Probability

The theoretical method gives the most reliable results, but it cannot always be used. If the sample space of an experiment consists of equally likely outcomes, then the theoretical probability of an event is defined to be the ratio of the number of outcomes in the event to the number of outcomes in the sample space.

For an experiment whose sample space S S consists of equally likely outcomes, the theoretical probability of the event E E is the ratio

P ( E ) = n ( E ) n ( S ) , P ( E ) = n ( E ) n ( S ) ,

where n ( E ) n ( E ) and n ( S ) n ( S ) denote the number of outcomes in the event and in the sample space, respectively.

Example 7.17

Computing theoretical probabilities.

Recall that a standard deck of cards consists of 52 unique cards which are labeled with a rank (the whole numbers from 2 to 10, plus J, Q, K, and A) and a suit ( ♣ ♣ , ♢ ♢ , ♡ ♡ , or ♠ ♠ ). A standard deck is thoroughly shuffled, and you draw one card at random (so every card has an equal chance of being drawn). Find the theoretical probability of each of these events:

• The card is 10 ♠ 10 ♠ .
• The card is a ♡ ♡ .
• The card is a king (K).

There are 52 cards in the deck, so the sample space for each of these experiments has 52 elements. That will be the denominator for each of our probabilities.

• There is only one 10 ♠ 10 ♠ in the deck, so this event only has one outcome in it. Thus, P ( 10 ♠ ) = 1 52 P ( 10 ♠ ) = 1 52 .
• There are 13 ♡ s ♡ s in the deck, so P ( ♡ ) = 13 52 = 1 4 P ( ♡ ) = 13 52 = 1 4 .
• There are 4 cards of each rank in the deck, so P ( K ) = 4 52 = 1 13 P ( K ) = 4 52 = 1 13 .

Your Turn 7.17

It is critical that you make sure that every outcome in a sample space is equally likely before you compute theoretical probabilities!

Example 7.18

Using tables to find theoretical probabilities.

In the Basic Concepts of Probability, we were considering a Monopoly game where, if your sister rolled a sum of 4, 5, or 7 with 2 standard dice, you would win the game. What is the probability of this event? Use tables to determine your answer.

We should think of this experiment as occurring in two stages: (1) one die roll, then (2) another die roll. Even though these two stages will usually occur simultaneously in practice, since they’re independent, it’s okay to treat them separately.

Step 1: Since we have two independent stages, let’s create a table (Figure 7.27), which is probably the most efficient method for determining the sample space.

Now, each of the 36 ordered pairs in the table represent an equally likely outcome.

Step 2: To make our analysis easier, let’s replace each ordered pair with the sum (Figure 7.28).

Step 3: Since the event we’re interested in is the one consisting of rolls of 4, 5, or 7. Let’s shade those in (Figure 7.29).

Our event contains 13 outcomes, so the probability that your sister rolls a losing number is 13 36 13 36 .

Your Turn 7.18

Example 7.19, using tree diagrams to compute theoretical probability.

If you flip a fair coin 3 times, what is the probability of each event? Use a tree diagram to determine your answer

• You flip exactly 2 heads.
• You flip 2 consecutive heads at some point in the 3 flips.
• All 3 flips show the same result.

Let’s build a tree to identify the sample space (Figure 7.30).

The sample space is {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}, which has 8 elements.

• Flipping exactly 2 heads occurs three times (HHT, HTH, THH), so the probability is 3 8 3 8 .
• Flipping 2 consecutive heads at some point in the experiment happens 3 times: HHH, HHT, THH. So, the probability is 3 8 3 8 .
• There are 2 outcomes that all show the same result: HHH and TTT. So, the probability is 2 8 = 1 4 2 8 = 1 4 .

Your Turn 7.19

People in mathematics, gerolamo cardano.

The first known text that provided a systematic approach to probabilities was written in 1564 by Gerolamo Cardano (1501–1576). Cardano was a physician whose illegitimate birth closed many doors that would have otherwise been open to someone with a medical degree in 16th-century Italy. As a result, Cardano often turned to gambling to help ends meet. He was a remarkable mathematician, and he used his knowledge to gain an edge when playing at cards or dice. His 1564 work, titled Liber de ludo aleae (which translates as Book on Games of Chance ), summarized everything he knew about probability. Of course, if that book fell into the hands of those he played against, his advantage would disappear. That’s why he never allowed it to be published in his lifetime (it was eventually published in 1663). Cardano made other contributions to mathematics; he was the first person to publish the third degree analogue of the Quadratic Formula (though he didn’t discover it himself), and he popularized the use of negative numbers.

Method 2: Empirical Probability

Theoretical probabilities are precise, but they can’t be found in every situation. If the outcomes in the sample space are not equally likely, then we’re out of luck. Suppose you’re watching a baseball game, and your favorite player is about to step up to the plate. What is the probability that he will get a hit?

In this case, the sample space is {hit, not a hit}. That doesn’t mean that the probability of a hit is 1 2 1 2 , since those outcomes aren’t equally likely. The theoretical method simply can’t be used in this situation. Instead, we might look at the player’s statistics up to this point in the season, and see that he has 122 hits in 531 opportunities. So, we might think that the probability of a hit in the next plate appearance would be about 122 531 ≈ 0.23 122 531 ≈ 0.23 . When we use the outcomes of previous replications of an experiment to assign a probability to the next replication, we’re defining an empirical probability . Empirical probability is assigned using the outcomes of previous replications of an experiment by finding the ratio of the number of times in the previous replications the event occurred to the total number of previous replications.

Empirical probabilities aren’t exact, but when the number of previous replications is large, we expect them to be close. Also, if the previous runs of the experiment are not conducted under the exact set of circumstances as the one we’re interested in, the empirical probability is less reliable. For instance, in the case of our favorite baseball player, we might try to get a better estimate of the probability of a hit by looking only at his history against left- or right-handed pitchers (depending on the handedness of the pitcher he’s about to face).

Probability and Statistics

One of the broad uses of statistics is called statistical inference, where statisticians use collected data to make a guess (or inference) about the population the data were collected from. Nearly every tool that statisticians use for inference is based on probability. Not only is the method we just described for finding empirical probabilities one type of statistical inference, but some more advanced techniques in the field will give us an idea of how close that empirical probability might be to the actual probability!

Example 7.20

Finding empirical probabilities.

Assign an empirical probability to the following events:

• Jose is on the basketball court practicing his shots from the free throw line. He made 47 out of his last 80 attempts. What is the probability he makes his next shot?
• Amy is about to begin her morning commute. Over her last 60 commutes, she arrived at work 12 times in under half an hour. What is the probability that she arrives at work in 30 minutes or less?
• Felix is playing Yahtzee with his sister. Felix won 14 of the last 20 games he played against her. How likely is he to win this game?
• Since Jose made 47 out of his last 80 attempts, assign this event an empirical probability of 47 80 ≈ 59 % 47 80 ≈ 59 % .
• Amy completed the commute in under 30 minutes in 12 of the last 60 commutes, so we can estimate her probability of making it in under 30 minutes this time at 12 60 = 20 % 12 60 = 20 % .
• Since Felix has won 14 of the last 20 games, assign a probability for a win this time of 14 20 = 70 % 14 20 = 70 % .

Your Turn 7.20

Work it out, buffon’s needle.

A famous early question about probability (posed by Georges-Louis Leclerc, Comte de Buffon in the 18th century) had to do with the probability that a needle dropped on a floor finished with wooden slats would lay across one of the seams. If the distance between the slats is exactly the same length as the needle, then it can be shown using calculus that the probability that the needle crosses a seam is 2 π 2 π . Using toothpicks or matchsticks (or other uniformly long and narrow objects), assign an empirical probability to this experiment by drawing parallel lines on a large sheet of paper where the distance between the lines is equal to the length of your dropping object, then repeatedly dropping the objects and noting whether the object touches one of the lines. Once you have your empirical probability, take its reciprocal and multiply by 2. Is the result close to π π ?

Method 3: Subjective Probability

In cases where theoretical probability can’t be used and we don’t have prior experience to inform an empirical probability, we’re left with one option: using our instincts to guess at a subjective probability . A subjective probability is an assignment of a probability to an event using only one’s instincts.

Subjective probabilities are used in cases where an experiment can only be run once, or it hasn’t been run before. Because subjective probabilities may vary widely from person to person and they’re not based on any mathematical theory, we won’t give any examples. However, it’s important that we be able to identify a subjective probability when we see it; they will in general be far less accurate than empirical or theoretical probabilities.

Example 7.21

Distinguishing among theoretical, empirical, and subjective probabilities.

Classify each of the following probabilities as theoretical, empirical, or subjective.

• An eccentric billionaire is testing a brand new rocket system. He says there is a 15% chance of failure.
• With 4 seconds to go in a close basketball playoff game, the home team need 3 points to tie up the game and send it to overtime. A TV commentator says that team captain should take the final 3-point shot, because he has a 38% chance of making it (greater than every other player on the team).
• Felix is losing his Yahtzee game against his sister. He has one more chance to roll 2 dice; he’ll win the game if they both come up 4. The probability of this is about 2.8%.
• This experiment has never been run before, so the given probability is subjective.
• Presumably, the commentator has access to each player’s performance statistics over the entire season. So, the given probability is likely empirical.
• Rolling 2 dice results in a sample space with equally likely outcomes. This probability is theoretical. (We’ll learn how to calculate that probability later in this chapter.)

Your Turn 7.21

Benford’s law.

In 1938, Frank Benford published a paper (“The law of anomalous numbers,” in Proceedings of the American Philosophical Society ) with a surprising result about probabilities. If you have a list of numbers that spans at least a couple of orders of magnitude (meaning that if you divide the largest by the smallest, the result is at least 100), then the digits 1–9 are not equally likely to appear as the first digit of those numbers, as you might expect. Benford arrived at this conclusion using empirical probabilities; he found that 1 was about 6 times as likely to be the initial digit as 9 was!

New Probabilities from Old: Complements

One of the goals of the rest of this chapter is learning how to break down complicated probability calculations into easier probability calculations. We’ll look at the first of the tools we can use to accomplish this goal in this section; the rest will come later.

Given an event E E , the complement of E E (denoted E ′ E ′ ) is the collection of all of the outcomes that are not in E E . (This is language that is taken from set theory, which you can learn more about elsewhere in this text.) Since every outcome in the sample space either is or is not in E E , it follows that n ( E ) + n ( E ′ ) = n ( S ) n ( E ) + n ( E ′ ) = n ( S ) . So, if the outcomes in S S are equally likely, we can compute theoretical probabilities P ( E ) = n ( E ) n ( S ) P ( E ) = n ( E ) n ( S ) and P ( E ′ ) = n ( E ′ ) n ( S ) P ( E ′ ) = n ( E ′ ) n ( S ) . Then, adding these last two equations, we get

P ( E ) + P ( E ′ ) = n ( E ) n ( S ) + n ( E ′ ) n ( S ) = n ( E ) + n ( E ′ ) n ( S ) = n ( S ) n ( S ) = 1 P ( E ) + P ( E ′ ) = n ( E ) n ( S ) + n ( E ′ ) n ( S ) = n ( E ) + n ( E ′ ) n ( S ) = n ( S ) n ( S ) = 1

Thus, if we subtract P ( E ′ ) P ( E ′ ) from both sides, we can conclude that P ( E ) = 1 − P ( E ′ ) P ( E ) = 1 − P ( E ′ ) . Though we performed this calculation under the assumption that the outcomes in S S are all equally likely, the last equation is true in every situation.

P ( E ) = 1 − P ( E ′ ) P ( E ) = 1 − P ( E ′ )

How is this helpful? Sometimes it is easier to compute the probability that an event won’t happen than it is to compute the probability that it will . To apply this principle, it’s helpful to review some tricks for dealing with inequalities. If an event is defined in terms of an inequality, the complement will be defined in terms of the opposite inequality: Both the direction and the inclusivity will be reversed, as shown in the table below.

Example 7.22

Using the formula for complements to compute probabilities.

• If you roll a standard 6-sided die, what is the probability that the result will be a number greater than one?
• If you roll two standard 6-sided dice, what is the probability that the sum will be 10 or less?
• If you flip a fair coin 3 times, what is the probability that at least one flip will come up tails?
• Here, the sample space is {1, 2, 3, 4, 5, 6}. It’s easy enough to see that the probability in question is 5 6 5 6 , because there are 5 outcomes that fall into the event “roll a number greater than 1.” Let’s also apply our new formula to find that probability. Since E E is defined using the inequality roll > 1 roll > 1 , then E ′ E ′ is defined using roll ≤ 1 roll ≤ 1 . Since there’s only one outcome (1) in E ′ E ′ , we have P ( E ′ ) = 1 6 P ( E ′ ) = 1 6 . Thus, P ( E ) = 1 − P ( E ′ ) = 5 6 P ( E ) = 1 − P ( E ′ ) = 5 6 .

Here, the event E E is defined by the inequality sum ≤ 10 sum ≤ 10 . Thus, E ′ E ′ is defined by sum > 10 sum > 10 . There are three outcomes in E ′ E ′ : two 11s and one 12. Thus, P ( E ) = 1 − P ( E ′ ) = 1 − 3 36 = 11 12 P ( E ) = 1 − P ( E ′ ) = 1 − 3 36 = 11 12 .

• In Example 7.15, we found the sample space for this experiment consisted of these equally likely outcomes: {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}. Our event E E is defined by T ≥ 1 T ≥ 1 , so E ′ E ′ is defined by T < 1 T < 1 . The only outcome in E ′ E ′ is the first one on the list, where zero tails are flipped. So, P ( E ) = 1 − P ( E ′ ) = 1 − 1 8 = 7 8 P ( E ) = 1 − P ( E ′ ) = 1 − 1 8 = 7 8 .

Your Turn 7.22

Check your understanding, section 7.5 exercises.

For the following exercises, use the following table of the top 15 players by number of plate appearances (PA) in the 2019 Major League Baseball season to assign empirical probabilities to the given events. A plate appearance is a batter’s opportunity to try to get a hit. The other columns are runs scored (R), hits (H), doubles (2B), triples (3B), home runs (HR), walks (BB), and strike outs (SO).

Statistics Made Easy

10 Examples of Using Probability in Real Life

Probability refers to how likely an event is to occur.

Probability is used in all types of areas in real life including weather forecasting, sports betting, investing, and more.

The following examples share how probability is used in 10 real-life situations on a regular basis.

Example 1: Weather Forecasting

Perhaps the most common real life example of using probability is weather forecasting .

Probability is used by weather forecasters to assess how likely it is that there will be rain, snow, clouds, etc. on a given day in a certain area.

Forecasters will regularly say things like “there is an 80% chance of rain today between 2PM and 5PM” to indicate that there’s a high likelihood of rain during certain hours.

Example 2: Sports Betting

Probability is heavily used by sports betting companies to determine the odds they should set for certain teams to win certain games.

For example, a sports betting company may look at the current record of two teams and determine that team A has a 90% probability of winning while team B has just a 10% probability of winning.

Based on these probabilities, the company would offer a higher payout for people who bet on team B to win since it’s highly unlikely that team B will actually win.

Example 3: Politics

Political forecasters use probability to predict the chances that certain candidates will win various elections.

For example, a forecaster might say that candidate A has a 60% chance of winning, candidate B has a 20% chance of winning, candidate C has a 10% chance of winning, etc. to give voters an idea of how likely it is that each candidate will win.

Note : A real-life example of a site that uses probability to perform political forecasting is FiveThirtyEight .

Example 4: Sales Forecasting

Many retail companies use probability to predict the chances that they’ll sell a certain amount of goods in a given day, week, or month.

This allows the companies to predict how much inventory they’ll need. For example, a company might use a forecasting model that tells them the probability of selling at least 100 products on a certain day is 90%.

This means they’ll need to make sure they have at least 100 products on hand to sell (or preferably more) so they don’t run out.

Example 5: Health Insurance

Health insurance companies often use probability to determine how likely it is that certain individuals will spend a certain amount on healthcare each year.

For example, a company might use factors like age, existing medical conditions, current health status, etc. to determine that there’s a 90% probability that a certain individual will spend $10,000 or more on healthcare in a given year.

Individuals who are likely to spend more on healthcare will be charged higher premiums because the insurance company knows that they’ll be more expensive to insure.

Example 6: Grocery Store Staffing

Grocery stores often use probability to determine how many workers they should schedule to work on a given day.

For example, a grocery store may use a model that tells them there is a 75% chance that they’ll have more than 800 customers come into the store on a given day.

Based on this probability, they’ll schedule a certain amount of workers to be at the store on that day to handle that many customers.

Example 7: Natural Disasters

The environmental departments of countries often use probability to determine how likely it is that a natural disaster like a hurricane, tornado, earthquake, etc. will strike the country in a given year.

If the probability is quite high, then the department will make decisions about housing, resource allocation, etc. that will minimize the effects done by the natural disaster.

Example 8: Traffic

Ordinary people use probability every day when they decide to drive somewhere.

Based on the time of day, location in the city, weather conditions, etc. we all tend to make probability predictions about how bad traffic will be during a certain time.

For example, if you think there’s a 90% probability that traffic will be heavy from 4PM to 5:30PM in your area then you may decide to wait to drive somewhere during that time.

Example 9: Investing

Investors use probability to assess how likely it is that a certain investment will pay off.

For example, a given investor might determine that there is a 1% chance that the stock of company A will increase 100x during the upcoming year.

Based on this probability, the investor will decide how much of their net worth to invest in the stock.

Example 10: Card Games

Probability is routinely used by anyone who plays card games on a regular basis.

For example, professional poker players use probability to determine how likely it is that a certain hand of cards will win and this informs them on how much they should bet.

If a player knows that there is a high probability that they will win a certain hand based on their cards, they will be more likely to bet more money.

Conversely, if they think the probability that they’ll win is low then they may bet significantly less money.

Additional Resources

The following tutorials provide additional information about probability:

Examples of Using Conditional Probability in Real Life Probability vs. Proportion: What’s the Difference? Probability vs. Likelihood: What’s the Difference? Law of Total Probability: Definition & Examples

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Reflection on Statistics Learning Goals Essay

Introduction, descriptive statistics, correlation, hypothesis testing, future plans, further study: faq.

Statistical knowledge is important to both statisticians and non-statisticians (Broers, 2006). It is, therefore, recommended that people from all disciplines are given basic skills in statistics.

For this reason, I pursued this course to obtain quantitative skills to be applied and improved on in several ways. In this regard, I hoped to obtain knowledge in designing of experiments, collection and analysis of data, interpretation of results as well as drawing of conclusions (Broers, 2006).

Today I can proudly say that my learning objectives were well met. I now have a clear understanding of statistical applications. I know how to collect, organize and describe data. I can apply measures of central tendency such as mean, mode and median in data description.

I can also use measures of dispersion such as standard deviation and variance to describe data. In addition, I have clear knowledge of normal distributions as well as the conditions to be met for a distribution to be considered normal (Broers, 2006).

Through this course, am able to determine a relationship between two sets of data. I can identify dependent and independent variables and the kind of relationship that exist between them. I can tell if one variable has a causal effect on the other. Furthermore, am able to identify positive, negative and minimal correlations (Broers, 2006).

In most cases, it is difficult to collect data on a whole population (Chance, 2002). In this regard, I have obtained necessary skills in sampling techniques. I have skills to analyze data from a sample and make conclusions regarding the entire population by using statistical probabilities and test of hypothesis.

In hypothesis testing, one tries to establish whether the outcomes of a certain study are due to chance or identifiable cause (Chance, 2002). Knowledge in application of significance level, critical value, degrees of freedom and p-value is used. One has to be able to formulate the null and the alternative hypotheses (Chance, 2002).

I am now in a position to use t-test to determine if there are statistically significant differences between two sets of data. In this regard, I understand the required assumptions for t-test to be applied. I have a clear understanding of analysis of variance (ANOVA), both one-way and two-way. I, however, feel that more practice in all these applications will help me perfect my understanding (Chance, 2002).

This course has given me a clear understanding of the role of statistics in life. Statistics is the most used research tool in medicine, education, psychology, business and economics, among other fields (Rumsey, 2002). It helps in shaping people’s choices in their daily lives. For example, statistical findings can give a clear understanding of implications of some behaviors such as smoking and lead to corrective measures.

I have always wanted to build a strong career in research. After taking this course, I have now made up my minds to major in statistics. I wish to have advanced skills in statistics which will enable me handle and analyze large and complex research problems. In this case, the knowledge I have already obtained will give me a head start.

The objectives I hoped to attain by pursuing this cause have been well achieved. Am now able to design experiments as well as collect, analyze and interpret data. I can apply this knowledge in real life situations and draw conclusions that will help find solutions to some problems.

However, this course has given me energy to pursue further knowledge in statistics. For this reason, I intend to major in statistics to be in a better position to handle more complex research problems.

Broers, N. J. (2006). Learning goals: the primacy of statistical knowledge . Maastricht: Maastricht University.

Chance, B. L. (2002). Components of statistical thinking and implications for instruction and assessment. Journal of Statistics Education , 10(3): 15-19.

Rumsey, D. J. (2002). Statistical literacy as a goal for introductory statistics courses. Journal of Statistics Education , 10(3): 7-13

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