4.109 Repeat the experiment many times. Here is a probability distribution for a random variable X:

A single experiment generates a random value from this distribution. If the experiment is repeated many times, what will be the approximate proportion of times that the value is 4? Give a reason for your answer.

4.110 Repeat the experiment many times and take the mean. Here is a probability distribution for a random variable X:

A single experiment generates a random value from this distribution. If the experiment is repeated many times, what will be the approximate value of the mean of these random variables? Give a reason for your answer.

4.111 Work with a transformation. Here is a probability distribution for a random variable X:

1. Find the mean and the standard deviation of this distribution.

2. Let Y=4X−2. Use the rules for means and variances to find the mean and the standard deviation of the distribution of Y.

3. For part (b), give the rules that you used to find your answer.

4.112 A different transformation. Refer to the previous exercise. Now let Y=4X2−2.

1. Find the distribution of Y.

2. Find the mean and standard deviation for the distribution of Y.

3. Explain why the rules that you used for part (b) of the previous exercise do not work for this transformation.

4.113 Roll a pair of dice two times. Consider rolling a pair of fair dice two times (see Exercise 4.48, page 234). For each of the following pairs of events, tell whether they are disjoint, independent, or neither.

1. A={8 on the first roll}, B={8 or more on the first roll}.

2. A={8 on the first roll}, B={8 or more on the second roll}.

3. A={6 or less on the second roll}, B={6 or more on the first roll}.

4. A={4 or less on the second roll}, B={6 or less on the second roll}.

4.114 Find the probabilities. Refer to the previous exercise. Find the probabilities for each event.

4.115 Some probability distributions. Here is a probability distribution for a random variable X:

1. Find the mean and standard deviation for this distribution.

2. Construct a different probability distribution with the same possible values, the same mean, and a larger standard deviation. Show your work and report the standard deviation of your new distribution.

3. Construct a different probability distribution with the same possible values, the same mean, and a smaller standard deviation. Show your work and report the standard deviation of your new distribution.

4.116 A fair bet at craps. Almost all bets made at gambling casinos favor the house. In other words, the difference between the amount bet and the mean of the distribution of the payoff is a positive number. An exception is “taking the odds” at the game of craps, a bet that a player can make under certain circumstances. The bet becomes available when a shooter throws a 4, 5, 6, 8, 9, or 10 on the initial roll. This number is called the “point”; when a point is rolled, we say that a point has been established. If a 4 is the point, an odds bet can be made that wins if a 4 is rolled before a 7 is rolled. The probability of winning this bet is 1/3, and the same payoff for a $10 bet is $20 (that is, you keep the $10 you bet and you receive an additional $20). The same probability of winning and payoff apply for an odds bet on a 10. For an initial roll of 5 or 9, the odds bet has a winning probability of 2/5, and the payoff for a $10 bet is $15. Similarly, when the initial roll is 6 or 8, the odds bet has a winning probability of 5/11, and the payoff for a $10 bet is $12.

1. Find the mean of the payoff distribution for each of these bets.

2. Confirm that the bets are fair by showing that the difference between the amount bet and the mean of the distribution of the payoff is zero.

4.117 An interesting case of independence. Independent events are not always easy to identify. Toss two balanced coins independently. The four possible combinations of heads and tails in order each have probability 0.25. The events

may seem intuitively related. Show that P(B|A)=P(B), so that A and B are, in fact, independent.

4.118 Lottery tickets. Michael buys a ticket in the Tri-State Pick 3 lottery every day, always betting on 491. He will win something if the winning number contains 4, 9, and 1 in any order. Each day, Michael has probability 0.006 of winning, and he wins (or not) independently of other days because a new drawing is held each day. What is the probability that Michael’s first winning ticket comes on the fifth day?

4.119 Sample surveys for sensitive issues. It is difficult to conduct sample surveys on sensitive issues because many people will not answer questions if the answers might embarrass them. Randomized response is an effective way to guarantee anonymity while collecting information on topics such as student cheating or sexual behavior. Here is the idea. To ask a sample of students whether they have plagiarized a term paper while in college, have each student toss a coin in private. If the coin lands heads and they have not plagiarized, they are to answer No. Otherwise, they are to give Yes as their answer. Only the student knows whether the answer reflects the truth or just the coin toss, but the researchers can use a proper random sample with follow-up for nonresponse and other good sampling practices.

Suppose that, in fact, the probability is 0.3 that a randomly chosen student has plagiarized a paper. Draw a tree diagram in which the first stage is tossing the coin and the second is the truth about plagiarism. The outcome at the end of each branch is the answer given to the randomized-response question. What is the probability of a No answer in the randomized-response poll? If the probability of plagiarism were 0.2, what would be the probability of a No response on the poll? Now suppose that you get 39% No answers in a randomized-response poll of a large sample of students at your college. What do you estimate to be the percent of the population who have plagiarized a paper?

4.120 Find some conditional probabilities. Choose a point at random in the square with sides 0≤x≤1 and 0≤y≤1. This means that the probability that the point falls in any region within the square is the area of that region. Let X be the x coordinate and Y the y coordinate of the point chosen. Find the conditional probability P(Y<1/3|Y>X). (Hint: Sketch the square and the events Y<1/3 and Y>X.)