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his or her money, what is the probability that you will lose your

[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 ¬rst generation
with probability .4. What is the probability that the mutants
eventually take over? [Hint: See Exercise 4.]

[Ans. .]

6. Verify that the proof of formula 4.8 in the text is still correct
when p > 2 . Interpret formula 4.8 for this case.
7. Show that if p > 2 , and both parties have a substantial amount
of money, your probability of ruin is approximately 1/r A .
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 Jones™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
[Ans. .]

(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
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?
[Ans. .]

(b) You bet all your money each time?
[Ans. .]

12. Repeat Exercise 11 for the case of a fair game, where you have
probability 1 of winning.

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

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 ¬rst?

[Ans. .]

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,
Goldberg, S., Probability: An Introduction, 1960.
Mosteller, F., Fifty Challenging Problems in Probability with Solutions,
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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