стр. 20 |

dollar?

[Ans. .868.]

2. Suppose that you are shooting craps, and you always hold the

dice. You have $20, your opponent has $10, and $1 is bet on each

game; estimate your probability of ruin.

3. Two government agencies, A and B, are competing for the same

task. A has 50 positions, and B has 20. Each year one position

is taken away from one of the agencies, and given to the other.

If 52 per cent of the time the shift is from A to B, what do you

predict for the future of the two agencies?

[Ans. One agency will be abolished. B survives with probability

.8, A with probability .2.]

4. What is the approximate value of xA if you are rich, and the

gambler starts with $1?

5. Consider a simple model for evolution. On a small island there is

room for 1000 members of a certain species. One year a favorable

mutant appears. We assume that in each subsequent generation

either the mutants take one place from the regular members of

the species, with probability .6, or the reverse happens. Thus,

for example, the mutation disappears in the very п¬Ѓrst generation

with probability .4. What is the probability that the mutants

eventually take over? [Hint: See Exercise 4.]

189

4.15. GAMBLERвЂ™S RUIN

1

[Ans. .]

3

6. Verify that the proof of formula 4.8 in the text is still correct

1

when p > 2 . Interpret formula 4.8 for this case.

1

7. Show that if p > 2 , and both parties have a substantial amount

of money, your probability of ruin is approximately 1/r A .

1

8. Modify the proof in the text to apply to the case p = 2 . What is

the probability of your ruin?

[Ans. B/N .]

9. You are matching pennies. You have 25 pennies to start with,

and your opponent has 35. What is the probability that you will

win all his or her pennies?

10. Jones lives on a short street, about 100 steps long. At one end of

the street is JonesвЂ™s home, at the other a lake, and in the middle

a bar. One evening Jones leaves the bar in a state of intoxication,

and starts to walk at random. What is the probability that Jones

will fall into the lake if

(a) Jones is just as likely to take a step to the right as to the

left?

1

[Ans. .]

2

(b) Jones has probability .51 of taking a step towards home?

[Ans. .119.]

11. You are in the following hopeless situation: You are playing a

1

game in which you have only 3 chance of winning. You have

$1, and your opponent has $7. What is the probability of your

winning all his or her money if

(a) You bet $1 each time?

1

[Ans. .]

255

(b) You bet all your money each time?

1

[Ans. .]

27

190 CHAPTER 4. PROBABILITY THEORY

12. Repeat Exercise 11 for the case of a fair game, where you have

probability 1 of winning.

2

13. Modify the proof in the text to compute yi , the probability of

reaching state N = 5.

14. Verify, in Exercise 13, that xi + yi = 1 for every state. Interpret.

Note: The following exercises deal with the following ruin prob-

lem: A and B play a game in which A has probability W of

winning. They keep playing until either A has won six times or

B has won three times.

15. Set up the process as a Markov chain whose states are (a, b),

where a is the number of times A won, and b the number of B

wins.

16. For each state compute the probability of A winning from that

position. [Hint: Work from higher a- and b-values to lower ones.]

17. What is the probability that A reaches his or her goal п¬Ѓrst?

1024

[Ans. .]

2187

18. Suppose that payments are made as follows: If A wins six games,

A receives $1, if B wins three games then A pays $1. What is the

expected value of the payment, to the nearest penny?

Suggested reading.

Cramer, Harald, The Elements of Probability Theory, Part I, 1955.

Feller, W., An Introduction to Probability Theory and its Applications,

1950.

Goldberg, S., Probability: An Introduction, 1960.

Mosteller, F., Fifty Challenging Problems in Probability with Solutions,

1965.

Neyman, J., First Course in Probability and Statistics, 1950.

Parzen, E., Modern Probability Theory and Its Applications, 1960.

Whitworth, W. A., Choice and Chance, with 1000 Exercises, 1934.

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