probability homework 2

Probability Calculator

Probability of two events.

To find out the union, intersection, and other related probabilities of two independent events.

Probability Solver for Two Events

Please provide any 2 values below to calculate the rest probabilities of two independent events.

Probability of a Series of Independent Events

Probability of a normal distribution.

Use the calculator below to find the area P shown in the normal distribution, as well as the confidence intervals for a range of confidence levels.

Related Standard Deviation Calculator | Sample Size Calculator | Statistics Calculator

Probability is the measure of the likelihood of an event occurring. It is quantified as a number between 0 and 1, with 1 signifying certainty, and 0 signifying that the event cannot occur. It follows that the higher the probability of an event, the more certain it is that the event will occur. In its most general case, probability can be defined numerically as the number of desired outcomes divided by the total number of outcomes. This is further affected by whether the events being studied are independent, mutually exclusive, or conditional, among other things. The calculator provided computes the probability that an event A or B does not occur, the probability A and/or B occur when they are not mutually exclusive, the probability that both event A and B occur, and the probability that either event A or event B occurs, but not both.

Complement of A and B

Given a probability A , denoted by P(A) , it is simple to calculate the complement, or the probability that the event described by P(A) does not occur, P(A') . If, for example, P(A) = 0.65 represents the probability that Bob does not do his homework, his teacher Sally can predict the probability that Bob does his homework as follows:

P(A') = 1 - P(A) = 1 - 0.65 = 0.35

Given this scenario, there is, therefore, a 35% chance that Bob does his homework. Any P(B') would be calculated in the same manner, and it is worth noting that in the calculator above, can be independent; i.e. if P(A) = 0.65, P(B) does not necessarily have to equal 0.35 , and can equal 0.30 or some other number.

Intersection of A and B

The intersection of events A and B , written as P(A ∩ B) or P(A AND B) is the joint probability of at least two events, shown below in a Venn diagram. In the case where A and B are mutually exclusive events, P(A ∩ B) = 0 . Consider the probability of rolling a 4 and 6 on a single roll of a die; it is not possible. These events would therefore be considered mutually exclusive. Computing P(A ∩ B) is simple if the events are independent. In this case, the probabilities of events A and B are multiplied. To find the probability that two separate rolls of a die result in 6 each time:

The calculator provided considers the case where the probabilities are independent. Calculating the probability is slightly more involved when the events are dependent, and involves an understanding of conditional probability, or the probability of event A given that event B has occurred, P(A|B) . Take the example of a bag of 10 marbles, 7 of which are black, and 3 of which are blue. Calculate the probability of drawing a black marble if a blue marble has been withdrawn without replacement (the blue marble is removed from the bag, reducing the total number of marbles in the bag):

Probability of drawing a blue marble:

P(A) = 3/10

Probability of drawing a black marble:

P(B) = 7/10

Probability of drawing a black marble given that a blue marble was drawn:

P(B|A) = 7/9

As can be seen, the probability that a black marble is drawn is affected by any previous event where a black or blue marble was drawn without replacement. Thus, if a person wanted to determine the probability of withdrawing a blue and then black marble from the bag:

Probability of drawing a blue and then black marble using the probabilities calculated above:

P(A ∩ B) = P(A) × P(B|A) = (3/10) × (7/9) = 0.2333

Union of A and B

In probability, the union of events, P(A U B) , essentially involves the condition where any or all of the events being considered occur, shown in the Venn diagram below. Note that P(A U B) can also be written as P(A OR B) . In this case, the "inclusive OR" is being used. This means that while at least one of the conditions within the union must hold true, all conditions can be simultaneously true. There are two cases for the union of events; the events are either mutually exclusive, or the events are not mutually exclusive. In the case where the events are mutually exclusive, the calculation of the probability is simpler:

A basic example of mutually exclusive events would be the rolling of a dice, where event A is the probability that an even number is rolled, and event B is the probability that an odd number is rolled. It is clear in this case that the events are mutually exclusive since a number cannot be both even and odd, so P(A U B) would be 3/6 + 3/6 = 1 , since a standard dice only has odd and even numbers.

The calculator above computes the other case, where the events A and B are not mutually exclusive. In this case:

Using the example of rolling dice again, find the probability that an even number or a number that is a multiple of 3 is rolled. Here the set is represented by the 6 values of the dice, written as:

Exclusive OR of A and B

Another possible scenario that the calculator above computes is P(A XOR B) , shown in the Venn diagram below. The "Exclusive OR" operation is defined as the event that A or B occurs, but not simultaneously. The equation is as follows:

As an example, imagine it is Halloween, and two buckets of candy are set outside the house, one containing Snickers, and the other containing Reese's. Multiple flashing neon signs are placed around the buckets of candy insisting that each trick-or-treater only takes one Snickers OR Reese's but not both! It is unlikely, however, that every child adheres to the flashing neon signs. Given a probability of Reese's being chosen as P(A) = 0.65 , or Snickers being chosen with P(B) = 0.349 , and a P(unlikely) = 0.001 that a child exercises restraint while considering the detriments of a potential future cavity, calculate the probability that Snickers or Reese's is chosen, but not both:

0.65 + 0.349 - 2 × 0.65 × 0.349 = 0.999 - 0.4537 = 0.5453

Therefore, there is a 54.53% chance that Snickers or Reese's is chosen, but not both.

Normal Distribution

The normal distribution or Gaussian distribution is a continuous probability distribution that follows the function of:

where μ is the mean and σ 2 is the variance. Note that standard deviation is typically denoted as σ . Also, in the special case where μ = 0 and σ = 1 , the distribution is referred to as a standard normal distribution. Above, along with the calculator, is a diagram of a typical normal distribution curve.

The normal distribution is often used to describe and approximate any variable that tends to cluster around the mean, for example, the heights of male students in a college, the leaf sizes on a tree, the scores of a test, etc. Use the "Normal Distribution" calculator above to determine the probability of an event with a normal distribution lying between two given values (i.e. P in the diagram above); for example, the probability of the height of a male student is between 5 and 6 feet in a college. Finding P as shown in the above diagram involves standardizing the two desired values to a z-score by subtracting the given mean and dividing by the standard deviation, as well as using a Z-table to find probabilities for Z. If, for example, it is desired to find the probability that a student at a university has a height between 60 inches and 72 inches tall given a mean of 68 inches tall with a standard deviation of 4 inches, 60 and 72 inches would be standardized as such:

Given μ = 68; σ = 4 (60 - 68)/4 = -8/4 = -2 (72 - 68)/4 = 4/4 = 1

The graph above illustrates the area of interest in the normal distribution. In order to determine the probability represented by the shaded area of the graph, use the standard normal Z-table provided at the bottom of the page. Note that there are different types of standard normal Z-tables. The table below provides the probability that a statistic is between 0 and Z, where 0 is the mean in the standard normal distribution. There are also Z-tables that provide the probabilities left or right of Z, both of which can be used to calculate the desired probability by subtracting the relevant values.

For this example, to determine the probability of a value between 0 and 2, find 2 in the first column of the table, since this table by definition provides probabilities between the mean (which is 0 in the standard normal distribution) and the number of choices, in this case, 2. Note that since the value in question is 2.0, the table is read by lining up the 2 row with the 0 column, and reading the value therein. If, instead, the value in question were 2.11, the 2.1 row would be matched with the 0.01 column and the value would be 0.48257. Also, note that even though the actual value of interest is -2 on the graph, the table only provides positive values. Since the normal distribution is symmetrical, only the displacement is important, and a displacement of 0 to -2 or 0 to 2 is the same, and will have the same area under the curve. Thus, the probability of a value falling between 0 and 2 is 0.47725 , while a value between 0 and 1 has a probability of 0.34134. Since the desired area is between -2 and 1, the probabilities are added to yield 0.81859, or approximately 81.859%. Returning to the example, this means that there is an 81.859% chance in this case that a male student at the given university has a height between 60 and 72 inches.

The calculator also provides a table of confidence intervals for various confidence levels. Refer to the Sample Size Calculator for Proportions for a more detailed explanation of confidence intervals and levels. Briefly, a confidence interval is a way of estimating a population parameter that provides an interval of the parameter rather than a single value. A confidence interval is always qualified by a confidence level, usually expressed as a percentage such as 95%. It is an indicator of the reliability of the estimate.

The relative frequency shows the proportion of data points that have each value. The frequency tells the number of data points that have each value.

Answers will vary. One possible histogram is shown below.

Find the midpoint for each class. These will be graphed on the x -axis. The frequency values will be graphed on the y -axis values.

• The 40 th percentile is 37 years.
• The 78 th percentile is 70 years.

Jesse graduated 37 th out of a class of 180 students. There are 180 – 37 = 143 students ranked below Jesse. There is one rank of 37.

x = 143 and y = 1. x + .5 y n x + .5 y n (100) = 143 + .5 ( 1 ) 180 143 + .5 ( 1 ) 180 (100) = 79.72. Jesse’s rank of 37 puts him at the 80 th percentile.

• For runners in a race, it is more desirable to have a high percentile for speed. A high percentile means a higher speed, which is faster.
• 40 percent of runners ran at speeds of 7.5 miles per hour or less (slower), and 60 percent of runners ran at speeds of 7.5 miles per hour or more (faster).

When waiting in line at the DMV, the 85 th percentile would be a long wait time compared to the other people waiting. 85 percent of people had shorter wait times than Mina. In this context, Mina would prefer a wait time corresponding to a lower percentile. 85 percent of people at the DMV waited 32 minutes or less. 15 percent of people at the DMV waited 32 minutes or longer.

The manufacturer and the consumer would be upset. This is a large repair cost for the damages, compared to the other cars in the sample. INTERPRETATION: 90 percent of the crash-tested cars had damage repair costs of $1,700 or less; only 10 percent had damage repair costs of $1,700 or more.

You can afford 34 percent of houses. 66 percent of the houses are too expensive for your budget. INTERPRETATION: 34 percent of houses cost $240,000 or less; 66 percent of houses cost $240,000 or more.

More than 25 percent of salespersons sell four cars in a typical week. You can see this concentration in the box plot because the first quartile is equal to the median. The top 25 percent and the bottom 25 percent are spread out evenly; the whiskers have the same length.

Mean: 16 + 17 + 19 + 20 + 20 + 21 + 23 + 24 + 25 + 25 + 25 + 26 + 26 + 27 + 27 + 27 + 28 + 29 + 30 + 32 + 33 + 33 + 34 + 35 + 37 + 39 + 40 = 738;

738 27 738 27 = 27.33

The most frequent lengths are 25 and 27, which occur three times. Mode = 25, 27

The data are symmetrical. The median is 3, and the mean is 2.85. They are close, and the mode lies close to the middle of the data, so the data are symmetrical.

The data are skewed right. The median is 87.5, and the mean is 88.2. Even though they are close, the mode lies to the left of the middle of the data, and there are many more instances of 87 than any other number, so the data are skewed right.

When the data are symmetrical, the mean and median are close or the same.

The distribution is skewed right because it looks pulled out to the right.

The mean is 4.1 and is slightly greater than the median, which is 4.

The mode and the median are the same. In this case, both 5.

The distribution is skewed left because it looks pulled out to the left.

Both the mean and the median are 6.

The mode is 12, the median is 13.5, and the mean is 15.1. The mean is the largest.

The mean tends to reflect skewing the most because it is affected the most by outliers.

sampling variability

induced variability

measurement variability

natural variability

For Fredo: z = .158  –  .166 .012 .158  –  .166 .012 = –0.67.

For Karl: z = .177  –  .189 .015 .177  –  .189 .015 = –.8.

Fredo’s z score of –.67 is higher than Karl’s z score of –.8. For batting average, higher values are better, so Fredo has a better batting average compared to his team.

• s x = ∑ f m 2 n − x ¯ 2 = 193,157.45 30 − 79.5 2 = 10.88 s x = ∑ f m 2 n − x ¯ 2 = 193,157.45 30 − 79.5 2 = 10.88
• s x = ∑ f m 2 n − x ¯ 2 = 380,945.3 101 − 60.94 2 = 7.62 s x = ∑ f m 2 n − x ¯ 2 = 380,945.3 101 − 60.94 2 = 7.62
• s x = ∑ f m 2 n − x ¯ 2 = 440,051.5 86 − 70.66 2 = 11.14 s x = ∑ f m 2 n − x ¯ 2 = 440,051.5 86 − 70.66 2 = 11.14
• Number the entries in the table 1–51 (includes Washington, DC; numbered vertically)
• Arrow over to PRB
• Press 5:randInt(
• Enter 51,1,8)

Eight numbers are generated (use the right arrow key to scroll through the numbers). The numbers correspond to the numbered states (for this example: {47 21 9 23 51 13 25 4}. If any numbers are repeated, generate a different number by using 5:randInt(51,1)). Here, the states (and Washington DC) are {Arkansas, Washington DC, Idaho, Maryland, Michigan, Mississippi, Virginia, Wyoming}.

Corresponding percents are {30.1, 22.2, 26.5, 27.1, 30.9, 34.0, 26.0, 25.1}.

• See Table 2.89 and Table 2.90 .
• Both graphs have a single peak.
• Both graphs use class intervals with width equal to $50
• The couples graph has a class interval with no values
• It takes almost twice as many class intervals to display the data for couples
• Answers may vary. Possible answers include the following. The graphs are more similar than different because the overall patterns for the graphs are the same.
• Check student's solution.
• Both graphs have a single peak
• Both graphs display six class intervals
• Both graphs show the same general pattern
• Answers may vary. Possible answers include the following. Although the width of the class intervals for couples is double that of the class intervals for singles, the graphs are more similar than they are different.
• Answers may vary. Possible answers include the following. You are able to compare the graphs interval by interval. It is easier to compare the overall patterns with the new scale on the couples graph. Because a couple represents two individuals, the new scale leads to a more accurate comparison.
• Answers may vary. Possible answers include the following. Based on the histograms, it seems that spending does not vary much from singles to individuals who are part of a couple. The overall patterns are the same. The range of spending for couples is approximately double the range for individuals.

Answers will vary.

• 1 – (.02+.09+.19+.26+.18+.17+.02+.01) = .06
• .19+.26+.18 = .63
• Check student’s solution.

40 th percentile will fall between 30,000 and 40,000

80 th percentile will fall between 50,000 and 75,000

• more children; the left whisker shows that 25 percent of the population are children 17 and younger; the right whisker shows that 25 percent of the population are adults 50 and older, so adults 65 and over represent less than 25 percent
• 62.4 percent
• Answers will vary. Possible answer: State University conducted a survey to see how involved its students are in community service. The box plot shows the number of community service hours logged by participants over the past year.
• Because the first and second quartiles are close, the data in this quarter is very similar. There is not much variation in the values. The data in the third quarter is much more variable, or spread out. This is clear because the second quartile is so far away from the third quartile.
• Each box plot is spread out more in the greater values. Each plot is skewed to the right, so the ages of the top 50 percent of buyers are more variable than the ages of the lower 50 percent.
• The black sports car is most likely to have an outlier. It has the longest whisker.
• Comparing the median ages, younger people tend to buy the black sports car, while older people tend to buy the white sports car. However, this is not a rule, because there is so much variability in each data set.
• The second quarter has the smallest spread. There seems to be only a three-year difference between the first quartile and the median.
• The third quarter has the largest spread. There seems to be approximately a 14-year difference between the median and the third quartile.
• IQR ~ 17 years
• There is not enough information to tell. Each interval lies within a quarter, so we cannot tell exactly where the data in that quarter is are concentrated.
• The interval from 31 to 35 years has the fewest data values. Twenty-five percent of the values fall in the interval 38 to 41, and 25 percent fall between 41 and 64. Since 25 percent of values fall between 31 and 38, we know that fewer than 25 percent fall between 31 and 35.

the mean percentage, x ¯ = 1,328.65 50 = 26.75 x ¯ = 1,328.65 50 = 26.75

The median value is the middle value in the ordered list of data values. The median value of a set of 11 will be the sixth number in order. Six years will have totals at or below the median.

• mean = 1,809.3
• median = 1,812.5
• standard deviation = 151.2
• first quartile = 1,690
• third quartile = 1,935

Hint: think about the number of years covered by each time period and what happened to higher education during those periods.

For pianos, the cost of the piano is .4 standard deviations BELOW the mean. For guitars, the cost of the guitar is 0.25 standard deviations ABOVE the mean. For drums, the cost of the drum set is 1.0 standard deviations BELOW the mean. Of the three, the drums cost the lowest in comparison to the cost of other instruments of the same type. The guitar costs the most in comparison to the cost of other instruments of the same type.

• x ¯ = 23.32 x ¯ = 23.32
• Using the TI 83/84, we obtain a standard deviation of: s x = 12.95. s x = 12.95.
• The obesity rate of the United States is 10.58 percent higher than the average obesity rate.
• Since the standard deviation is 12.95, we see that 23.32 + 12.95 = 36.27 is the disease percentage that is one standard deviation from the mean. The U.S. disease rate is slightly less than one standard deviation from the mean. Therefore, we can assume that the United States, although 34 percent have the disease, does not have an unusually high percentage of people with the disease.
• For graph, check student's solution.
• 49.7 percent of the community is under the age of 35
• Based on the information in the table, graph (a) most closely represents the data.
• 174, 177, 178, 184, 185, 185, 185, 185, 188, 190, 200, 205, 205, 206, 210, 210, 210, 212, 212, 215, 215, 220, 223, 228, 230, 232, 241, 241, 242, 245, 247, 250, 250, 259, 260, 260, 265, 265, 270, 272, 273, 275, 276, 278, 280, 280, 285, 285, 286, 290, 290, 295, 302
• 205.5, 272.5
• .84 standard deviations below the mean

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Probability

How likely something is to happen.

Many events can't be predicted with total certainty. The best we can say is how likely they are to happen, using the idea of probability.

Tossing a Coin

When a coin is tossed, there are two possible outcomes:

Heads (H) or Tails (T)

• the probability of the coin landing H is ½
• the probability of the coin landing T is ½

Throwing Dice

When a single die is thrown, there are six possible outcomes: 1, 2, 3, 4, 5, 6 .

The probability of any one of them is 1 6

In general:

Probability of an event happening = Number of ways it can happen Total number of outcomes

Example: the chances of rolling a "4" with a die

Number of ways it can happen: 1 (there is only 1 face with a "4" on it)

Total number of outcomes: 6 (there are 6 faces altogether)

So the probability = 1 6

Example: there are 5 marbles in a bag: 4 are blue, and 1 is red. What is the probability that a blue marble gets picked?

Number of ways it can happen: 4 (there are 4 blues)

Total number of outcomes: 5 (there are 5 marbles in total)

So the probability = 4 5 = 0.8

Probability Line

We can show probability on a Probability Line :

Probability is always between 0 and 1

Probability is Just a Guide

Probability does not tell us exactly what will happen, it is just a guide

Example: toss a coin 100 times, how many Heads will come up?

Probability says that heads have a ½ chance, so we can expect 50 Heads .

But when we actually try it we might get 48 heads, or 55 heads ... or anything really, but in most cases it will be a number near 50.

Learn more at Probability Index .

Some words have special meaning in Probability:

Experiment : a repeatable procedure with a set of possible results.

Example: Throwing dice

We can throw the dice again and again, so it is repeatable.

The set of possible results from any single throw is {1, 2, 3, 4, 5, 6}

Outcome: A possible result.

Example: "6" is one of the outcomes of a throw of a die.

Trial: A single performance of an experiment.

Example: I conducted a coin toss experiment. After 4 trials I got these results:

Three trials had the outcome "Head", and one trial had the outcome "Tail"

Sample Space: all the possible outcomes of an experiment.

Example: choosing a card from a deck

There are 52 cards in a deck (not including Jokers)

So the Sample Space is all 52 possible cards : {Ace of Hearts, 2 of Hearts, etc... }

The Sample Space is made up of Sample Points:

Sample Point: just one of the possible outcomes

Example: Deck of Cards

• the 5 of Clubs is a sample point
• the King of Hearts is a sample point

"King" is not a sample point. There are 4 Kings, so that is 4 different sample points.

There are 6 different sample points in that sample space.

Event: one or more outcomes of an experiment

Example Events:

An event can be just one outcome:

• Getting a Tail when tossing a coin
• Rolling a "5"

An event can include more than one outcome:

• Choosing a "King" from a deck of cards (any of the 4 Kings)
• Rolling an "even number" (2, 4 or 6)

Hey, let's use those words, so you get used to them:

Example: Alex wants to see how many times a "double" comes up when throwing 2 dice.

The Sample Space is all possible Outcomes (36 Sample Points):

{1,1} {1,2} {1,3} {1,4} ... ... ... {6,3} {6,4} {6,5} {6,6}

The Event Alex is looking for is a "double", where both dice have the same number. It is made up of these 6 Sample Points :

{1,1} {2,2} {3,3} {4,4} {5,5} and {6,6}

These are Alex's Results:

 After 100 Trials , Alex has 19 "double" Events ... is that close to what you would expect?

Fall 2020, Lecture B00 (Kemp) TR 12:30-1:50pm

Probability theory i.

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Last Modified: 9/23/15

• Welcome : to Math 280A: Probability Theory I, in Fall 2020!
• Quiz 1 grades have been released on Gradescope. The regrade request window is now close.
• Quiz 2 grades have been released on Gradescope. The regrade request window is Monday 10/26/20 8am to Wednesday 10/28/20 at 8pm . If you wish to lodge a regrade request on Quiz 1, please do so during this period of time.

Course Information

• Textbook : The main source we will follow are Bruce Driver's excellent Probability notes: Probability Tools with Examples , by Bruce Driver Here are a few other textbooks we recommend as auxiliary sources; all are freely available to UCSD personnel. Probability: Theory and Examples (5th Edition), by Rick Durrett A Probability Path by Sidney Resnick A Modern Approach to Probability Theory by Bert Fristedt and Lawrence Gray Probability Theory: A Comprehensive Course by Achim Klenke If you are not on UCSD campus, make sure you are logged into the VPN in order to gain access; you can find instructions on how to do this here .
• Coursework : There will be weekly homework assignments due on Mondays (starting in Week 2); they are posted below . There will be 5 quizzes , in weeks 1, 3, 5, 7, and 9 of the quarter; they will take place during the scheduled Thursday lecture time, with an alternate sitting available in the late evening to accommodate those in distant time-zones. And there will be a take-home final exam during exam week. Timing and due dates for all courses assessments can be found below .
• Piazza is an online discussion forum. It will allow you to post messages (openly or anonymously) and answer posts made by your fellow students, about course content, homework, quizzes, etc. The instructor and TA will also monitor and post to Piazza regularly. You can sign up here . Note: Piazza has an opt-in "Piazza Careers" section which, if you give permission, will share statistics about your Piazza use with potential future employers. It also has a "social network" component, based on other students who've shared a Piazza-based class with you, that comes with the usual warnings about privacy concerns. Piazza is fully FERPA compliant, and is an allowed resource at UCSD. Nevertheless, you are not required to use Piazza if you do not wish.
• Gradescope is an online tool for uploading and grading assignments and exams (it is now under the umbrella of Turnitin). You will turn in your homework, quizzes, and final exam through Gradescope, and you will access your graded assessments there as well. Access the class Gradescope site here .

Instructional Staff

We will be communicating with you and making announcements through an online question and answer platform called Piazza (sign up link: piazza.com/ucsd/winter2016/math180a ). We ask that when you have a question about the class that might be relevant to other students, you post your question on Piazza instead of emailing us. That way, everyone can benefit from the response. Posts about homework or exams on Piazza should be content based. While you are encouraged to crowdsource and discuss coursework through Piazza, please do not post complete solutions to homework problems there. Questions about grades should be brought to the instructors, in office hours. You can also post private messages to instructors on Piazza, which we prefer to email.

Our office hours, and all relevant scheduled course activities, can be found in the following calendar.

Recorded Lectures

The Lectures for this course are pre-recorded, and available on YouTube .

The lectures are not divided into even 80-minute chunks. They are organized by topic, concept, or example.

Below, you will find a list (with links) of the lecture videos you should watch prior to the listed date, along with pdf slides of the tablet output during those lecture videos.

Math 280A is the first quarter of a three-quarter graduate level sequence in the theory of probability. This sequence provides a rigorous treatment of probability theory, using measure theory, and is essential preparation for Mathematics PhD students planning to do research in probability. A strong background in undergraduate real analysis at the level of Math 140AB is essential for success in Math 280A. In particular, students should be comfortable with notions such as countable and uncountable sets, limsup and liminf, and open, closed, and compact sets, and should be proficient at writing rigorous epsilon-delta style proofs. Graduate students who do not have this preparation are encouraged instead to consider Math 285, a one-quarter course in stochastic processes which will be offered in Winter 2021. See also this page , maintained by Ruth Williams, for more information on graduate courses in probability at UCSD.

According to the UC San Diego Course Catalog , the topics covered in the full-year sequence 280ABC include the measure-theoretic foundations of probability theory, independence, the Law of Large Numbers, convergence in distribution, the Central Limit Theorem, conditional expectation, martingales, Markov processes, and Brownian motion. Given the current pandemic crisis and emergency remote teaching modality, it is more difficult than usual to predict what pace we will work through this material, and where the dividing line between 280A and 280B will occur.

Prerequisite:   Students should have mastered the fundamentals of real analysis in metric spaces, as covered in MATH 140AB, before taking this course. An undergraduate course in probability, comparable to MATH 180A, and further courses in stochastic processes, comparable to MATH 180BC, would also be an asset, but are not absolutely necessary.

Lectures:   The lectures for this course will be recorded asynchronously, and made available on YouTube. You should engage with the relevant videos before each "Lecture" session. The schedule Lecture times will be devoted to Q&A sessions and quizzes. The Q&A sessions will be recorded, with recordings available on Canvas; the quizzes will not be recorded (but will take place live on Zoom), and will be available in a "second sitting" to accommodate those students in far-flung time-zones.

Homework:   Homework assignments are posted below , and will be due by 9pm (with a 30-minutes "late" grace period in case of technical glitches) on Mondays throughout thee quarter. You must turn in your homework through Gradescope; if you have produced it on paper, you can scan it or simply take clear photos of it to upload. You must select pages corresponding to your solutions of problems during the upload process. Gradescope will allow you to re-select pages at any point until grading has begun. If you have not selected pages when the TA begins grading, the TA will not grade your assignment and you will receive a grade of 0 on it. No appeals of this policy will be considered. It is allowed and even encouraged to discuss homework problems with your classmates and your instructor and TA, but your final write up of your homework solutions must be your own work.

Quizzes:   There will be 5 quizzes throughout the quarter, to test your fundamental knowledge of the course material. You will write them on Thursdays 1-1:50pm or 7-7:50pm, live on Zoom (so that your instructional team can answer questions if any arise), and turn them in via Gradescope. No collaboration (with other humans or with online resources) is allowed on quizzes.

Lowest scores:    Of the 9 homework assignments, only your highest 7 scores will count towards your final grade. Of the 5 quizzes, only your highest 4 scores will count towards your final grade.

Final Exam:   The final exam will be take-home. It will be available and due during exam week; more details about the exam window will be available later in the term. No collaboration (with other humans or with online resources) is allowed on the final exam. We reserve the right to invite students to follow-up Zoom meetings after the final exam to confirm that the work was completed without collaboration. We reserve the scheduled final exam time-slot for this purpose.

• Exam Responsibilities   An outline of the responsibilities of faculty and students with regard to final exams
• Policies on Examinations   The Academic Senate policy regarding final examinations (These are the rules!)

Regrade Policy:    Your quizzes, homeworks, and final exam will be graded using Gradescope . For quizzes and the final exam , you will be able to request regrades through Gradescope for a specified window of time. Be sure to make your request within the specified window of time; no regrade requests will be accepted after the deadline. For homework , any clerical erros (such as a problem or page that the TA accidentally missed when grading) should be discussed with the TA during office hours. Grading rubrics are not negotiable ; if the TA has taken off some number of points from your solution, there is a sound pegagogical reason for this. This is a PhD class in mathematics . We are not focused on numerical grades here; we are focused on learning deep and challenging material. The grading is meant as a formative assessment tool; if your grade is not perfect, it indicates you should spend more time reviewing the concepts and thinking about the problems. The TA will give detailed feedback in the grading; it is your responsibility to think and work hard to understand what concepts and ideas you need a firmer understanding of from any assignment where you did not receive full points. Only after working hard on your own, or in collaboration with fellow classmates (for example through Piazza), should you consider approaching your TA or instructor for further explanation of grading choices. However, please understand that these conversations will not result in a change in your grade unless there has been some clear clerical error, such as the TA accidentally missing part of your solution. The TA will not change their assessment of a students work due to conversations or complaints after the fact.

• 40% Homework,  30% Quizzes,  30% Final Exam

Academic Integrity:   UC San Diego's code of academic integrity outlines the expected academic honesty of all students and faculty, and details the consequences for academic dishonesty. The main issues are cheating and plagiarism, of course, for which we have a zero-tolerance policy. (Penalties for these offenses always include assignment of a failing grade in the course, and usually involve an administrative penalty, such as suspension or expulsion, as well.) However, academic integrity also includes things like giving credit where credit is due (listing your collaborators on homework assignments, noting books or papers containing information you used in solutions, etc.), and treating your peers and instructors respectfully in all forms of interaction (Zoom meetings, email, Piazza discussions, etc).

Assessment Versioning : following UCSD (and common) practice, recommended by the Academic Integrity Office, assessments given at non-overlapping times will be comparable , but may not be identical . In particular: the two sittings of each Quiz (and potentially quizzes given within each sitting) may not have exactly the same questions, but will be designed to cover the same material and be of equivalent levels of difficulty. This practice is meant to maintain course integrity, avoiding unpermitted collaboration (either intentional or accidental).

OSD Accommodations: Students requesting accommodations for this course due to a disability must provide a current Authorization for Accommodation (AFA) letter issued by the Office for Students with Disabilities (OSD) which is located in University Center 202 behind Center Hall. The AFA letter will be issued by the OSD electronically. Please make arrangements to discuss your accommodations with me in advance ( by the end of Week 2 ). We will make every effort to arrange for whatever accommodations are stipulated by your AFA letter. For more information, see here .

Covid-19 Accommodations: Due to the unprecedented Covid-19 pandemic, our lectures and all meetings will be remote, using Zoom, this year. Some of you are residing in different time-zones (and different continents), and these present additional challenges. To accommodate, all lectures are pre-recorded asynchronously, and the Q&A sessions are also recorded. Office hours will not be recorded, but will be distributed throughout different days and times so all students should be able to attend in regular work hours; failing that, individual Zoom meetings can be made. Synchronous quizzes will be held in two "sittings": 1-1:50pm and 7-7:50pm Pacific Time, which should cover everyone enrolled in the class, during daylight hours. If these accommodations are still insufficient due to a severe Covid-19 pandemic related issue you have, please contact me no later than Friday, October 16 , to discuss other arrangements.

Weekly homework assignments are posted here. Homework is due by 9:00pm on the posted date, through Gradescope . Late homework will not be accepted.

• Homework 1 , due Monday, October 12.
• Homework 2 , due Monday, October 19.
• Homework 3 , due Monday, October 26.
• Homework 4 , due Monday, November 2.
• Homework 5 , due Monday, November 9.
• Homework 6 , due Monday, November 16.
• Homework 7 , due Monday, November 23.
• Homework 8 , due Monday, November 30.
• Homework 9 , due Monday, December 7.
• Department of Mathematics, UC San Diego

BUS204: Business Statistics

BUS204 Study Guide

Unit 2: counting, probability, and probability distributions, 2a. identify values of and differentiate between permutations and combinations.

• What is the difference between a combination and a permutation?

It is not only the order that matters. If the selected members can be assigned any unique characteristics, such as 3 people being elected President, Vice President, and Treasurer from a club of 20 people, we are looking at a permutation, since for any group of 3 members, it does matter which title each person has, just like it would matter what order they'd be in if they stood in a line.

For more outside of what we've already covered, see Permutations and Combinations .

2b. Explain and apply the different methods for determining probability: equally likely outcomes, frequency theory, and subjective theory

• Define probability. 
• Describe the method of equally likely outcomes and how it can be used to find probabilities.

An outcome is a single possible result for an experiment. An event is made up of several outcomes. For example, {1,2,3,4,5,6} are the possible outcomes for a die roll. The event "Roll greater than a 4" is made of the outcomes {5,6}.

The set of possible outcomes for a probability experiment is called the sample space. A set is made of elements.

The fundamental definition of probability is the number of possible "successful" events or outcomes divided by the total number of possible events or outcomes. So the probability of rolling a 5 on a 6-sided die is 1/6 because the "success" is rolling a 5 (one possible success) out of 6 possible outcomes. We can use this principle as long as each outcome is equally likely.

To review, see Terminology .

2c. Define and apply the axioms of probability theory

• What is a compound event? How do unions and intersections of events relate to unions and intersections of sets?
• What is the difference between independent and dependent events?
• What is the difference between mutually exclusive and non-mutually exclusive events?

The addition rule is used to find the probability of an "or" compound event. 

The multiplication rule is used to find "and" probabilities:

To review, see Independent and Mutually Exclusive Events , Two Basic Rules of Probability , and Venn Diagrams , and see Addition Rule for Probability .

2d. Apply probability distributions and explain the properties of different distributions

• What is a random variable? What is the difference between a discrete and a continuous random variable?
• How is a probability distribution similar or different from a relative frequency distribution?
• What are some uses for probability distributions?
• What is sampling error, and why does it occur? Does it imply a mistake in data gathering?

Remember: a random variable is a variable that has its value as a result of an experiment or survey as opposed to solving an equation. A discrete random variable has a fixed number of possible values. A continuous random variable cannot easily have all its possible values listed.

A probability distribution consists of each possible value (or interval of values) of a random variable and the probability that the variable will take on that value. Probability distributions have many implications in decision-making. When you interpret data, you will need to know what probability distribution the value you're trying to estimate follows.

As an example, if you think that 30% of consumers might be interested in a particular product, and sample data from a small group shows 26%, you would conduct a statistical test (see Unit 5) what the probability is that a confidence interval of sample proportions would include 26% if the true value is 30%. Remember, even if the true population proportion is 30%, because of random sampling error, the proportions of samples will likely differ from 30%, so we have to find if the 26% is a significant enough difference to cast doubt on the hypothesis of 30%.

Sampling error occurs when conducting statistical tests because taking samples of a population will result in different sample means since each sample is unique. For example, if the mean of a population is 10, then the sample means should at least cluster around 10, but may not be exactly 10.

To review, see Probability Density Functions and Random Variables .

2e. Solve problems using the binomial distribution, and explain when it should be used

• What properties must be true of all the experiments in an event to make that a binomial event?
• When calculating binomial probabilities, why is it important to use the formula for calculating the number of combinations?
• Why is it important that the probabilities of success be equal for each experiment?
• Why is it important that the experiments are independent of each other?

A binomial event has the following properties:

Each experiment will result in either a "success" or "failure" (two possible outcomes). Success is simply defined as the result we are looking for, whether it is a good or bad occurrence.

To review, see Binomial Distribution .

2f. Differentiate between discrete and continuous probability distributions

• What is the difference between a discrete and a continuous distribution?
• Is the Binomial Distribution considered discrete or continuous?

Discrete probability distributions can have all possible outcomes listed. The random variable is discrete.

Continuous probability distributions are made up of possible intervals of values for a continuous random variable, and the individual values of the random value cannot all be listed – but the intervals they're grouped into can be.

Examples of discrete distributions are uniform-discrete, binomial, and Poisson distributions. Examples of continuous distributions are uniform-continuous and normal distributions.

2g. Apply expected value and calculate it for various probability distributions

• Why is the mean of a distribution often referred to as its expected value?
• If a set of 20 binomial experiments are continuously run, where each experiment has a 0.25 probability of success, and sets of 20 experiments are performed continuously and the number of successes out of 20 is recorded, what should we expect those numbers to average (arithmetic mean) to?

To review, see Terminology . This passage refers to the long-term probability of an event. Expected value is the long-term mean of random variables occurring with that probability.

Unit 2 Vocabulary

This vocabulary list includes terms that might help you with the review items above and some terms you should be familiar with to be successful in completing the final exam for the course. 

Try to think of the reason why each term is included.

• addition rule
• binomial random variable
• combination
• compound events
• conditional probability
• continuous probability distribution
• continuous random variable
• discrete probability distribution
• discrete random variable
• expected value
• independent events
• intersection
• multiplication rule
• mutually exclusive events
• permutation
• probability
• probability distribution
• random variable
• sampling error

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Want additional help with Algebra 2?

Unit 1: Polynomial arithmetic

Unit 2: complex numbers, unit 3: polynomial factorization, unit 4: polynomial division, unit 5: polynomial graphs, unit 6: rational exponents and radicals, unit 7: exponential models, unit 8: logarithms, unit 9: transformations of functions, unit 10: equations, unit 11: trigonometry, unit 12: modeling.

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