Statistics 101: Problem Set #2

1. 1. Page 88 Exercise 3.15

   2. Show: If A is not independent of B then A complement is not independent of B complement

   3. Two tennis players A and B play two tennis matches against each other. The probability that A wins the first match is .5. The probability that A wins the second match is also .5. But the probability that A wins the second match given A wins the first match is .6. What is the probability that player A wins at least one match?

2. 1. Page 100 Exercise 3.28

   2. Page 100 Exercise 3.29

3. Three marksmen A, B, and C engage in a contest. The marksmen who is alive last wins 1 million dollars. Each marksman gets to shoot in turn at one of the other two marksmen (i.e., A can shoot at B or C; B at A or C; C at A or B). Each marksman has two bullets. A has a .3 chance of hitting a target, B has a .5 chance, and C always hits the target. They go sequentially, with A going first, B second (if he is still alive), C third (if he still alive), then back to A, B, and finally C. At any point a marksman can throw his bullet away (i.e., shoot in the air). If more than one marksman is alive then no one wins.
   1. Show that it is best for A to throw his first bullet away.

   2. Find the optimal strategy for each marksman to maximize the probability of winning the 1 million dollars.

   3. If all of the marksmen follow optimal strategies, what are the probabilities that A, B, and C win?

4. 1. Page 112 Exercise R13

   2. Page 112 Exercise R14

5. 1. Page 143 Exercise 4.37

   2. Page 143 Exercise 4.38

   3. Page 143 Exercise 4.39

6. You go to Atlantic City with $32. You decide to play roulette and always bet on Red. Note that the true probability of Red is 18/38, but assume it is ½ to simplify the calculations.

   Strategy 1: You bet $8 on Red on four consecutive rolls. Let Y₁ = amount you go home with under strategy 1.

   1. Find the discrete probability distribution and cumulative distribution function for Y₁ .

   Strategy 2: You bet $1 on Red on the first roll. If you win you go home. If you lose, you bet $2 on Red on the second roll. If you win you go home. If you lose, you bet $4 on Red on the third roll. If you win you go home. If you lose, you bet $8 on Red on the fourth roll. If you win you go home. If you lose, you make one more bet of $16 on Red on the fifth roll. You go home after the fifth roll.

   2. Find the discrete probability distribution and cumulative distribution function for Y₂ .
   3. Devise two criteria for comparing the above two strategies and indicate which Strategy is best for each of the criterions.

Suggested Problems (not to be turned in):

• Page 83- Exercises 3.3; 3.4; 3.5; 3.6; 3.7
• Page 87- Exercise 3.13
• Page 99- Exercise 3.27
• Page 104- Exercise 3.44; 3.45
• Page 124- Exercise 4.8; 4.9
• Page 142- Exercise 4.32; 4.33; 4.34