Stat 20 Review Faculty: Professor Hank Ibser Reviewer: Larry Wang, [email protected] Location: 170 Barrows, Tu 5-7 PM Community through Academics and Leadership Study group: MW 12-1 201A Chavez , http://www.csrjjsmp.com/stat20.html

Midterm 2 Review 1) P(BC)=0.4 P(B|AC)=0.5 P(A|B)=0.6 Find the following probabilities: a) P(A) b) P(AC|BC) c) P(BC and A) d) P(A or B)

2) A cup contains 6 coins, 2 of which are fair, and 4 of which are weighted to land heads with probability 2/3. A coin is removed from the cup and tossed twice. The first time it lands heads, and the second time it lands tails. Based on this information, what is the probability that it is a fair coin?

3) A random variable X takes on a value according to the following table: x 2 4 5 7 9 P(X=x) 1/8 3/8 1/4 1/8 1/8 Y has the same expected value and standard error as X, but we know nothing about its distribution or whether it depends on X. Find: a) E(X2) e) E(3X-2Y-1) b) E(Y) f) SE(X+X) c) E(2X+1) g) E[(Y+1)2] d) SE(1.5X+2) h) SE(X+2Y)

4) A casino wants to introduce a new blackjack based game. A player gets to draw two cards from a standard deck. If the cards are an ace and a card that is worth 10 points in blackjack (10, J, Q, K), the player wins 20 dollars. If the player draws an ace and a black Jack, he wins 200 dollars instead. How much does the casino need to charge for this game so that they can avoid losing money?

5) 5 friends, Al, Beatrice, Cassie, Danielle, and Evan, all put 3 copies each of their names in a hat to be drawn out with replacement. a. Al will draw 12 times from the hat. What is the expected number of times he draws his own name? b. Beatrice will draw 10 times from the hat. How many vowels is she likely to see? Give or take how many? c. Cassie will draw 30 times from the hat. What is the chance that at least 22 of the names she draws have an “e” in them? d. Danielle will draw 30 times as well. What is the chance that the number of letters she draws is between 180 and 200, inclusive? e. Evan arrived late to the name-drawing party, and his friends have taken the opportunity to stuff the hat with more copies of their own names. He draws 100 times, and is surprised to find that he has only drawn his own name 3 times. Can he be 90% sure his name does not appear in the box as much as it should? Stat 20 Review Faculty: Professor Hank Ibser Reviewer: Larry Wang, [email protected] Location: 170 Barrows, Tu 5-7 PM Community through Academics and Leadership Study group: MW 12-1 201A Chavez , http://www.csrjjsmp.com/stat20.html

6) For each pair of events, determine which is more likely: a. Flipping 150 coins and getting more than 80 heads vs. flipping 200 coins and getting more than 105 heads. b. Flipping 150 coins and getting more than 60 heads vs. flipping 300 coins and getting more than 135 heads. c. Flipping 100 coins and getting more than 58 heads vs. flipping 400 coins and getting fewer than 180 heads.

7) A simple random sample of 49 households is conducted from a town with 674 houses. The head of the household is given a survey. Compiling the results, it is found that finds that there are 75% of households that have cable, 50% of the people living in those 49 houses are female, and 1% of the households don’t have running water. If possible, calculate the 95% confidence interval based on each of these data for the town. If any are not possible, explain what the problem is.

8) Say whether each of the statements are true or false: a. An investigator conducts a survey by waiting on a street corner and rolling a die whenever a person walks by. If the die comes up 5, he will interview the person. Since he is using a random event as the basis for his decision, this is a simple random sample. b. A simple random sample of adults in a certain town is collected. Each person will be asked a series of questions. If asking the questions in a certain order is likely to change the answers, this is an example of response bias. c. An investigator wants to estimate the average class size at Berkeley. To do so, he goes through the course catalog, and averages the size of every 25th class. Because the classes in the catalog are not arranged by class size, this is essentially random, and makes a probability sample.

9) You have a bag of marbles with 100 blue marbles, 36 red marbles, and 64 green marbles. Suppose you draw 64 marbles with replacement. a. How many marbles do you expect to be green? Attach a “give or take” number to this. b. What percent do you expect to be green? Attach a give or take number to this. c. What is the chance of having more than 40% green marbles?

10) Suppose that when you tell a person to show up at noon, the actual time they show up has a normal distribution with average noon but with an SD of 15 minutes. You tell thirty people to show up at noon. Assume they all arrive independently of each other and that time is continuous. a. What is the chance that all thirty are early? b. What is the chance that they all arrive before 12:30? c. What is the chance that the earliest person arrived before 11:30? d. What is the chance that the earliest person came before 11:30, given that they were all there by noon? e. What is the chance that they will all be there by noon, given that you know that the earliest one came before 11:30?