The Probability of Winning with Timid Play
The Probability of Winning with Timid Play |
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Interactive red and black game
Simulation of the red and black experiment
Recall that with timid play, the gambler makes a small constant bet, say $1, on each trial until he stops. Thus, on each trial, the gambler's fortune either increases by 1 or decreases by 1, until the fortune reaches either 0 or the target a (a positive integer). Thus, the fortune process
Xi, i = 0, 1, 2, ...
is a random walk with 0 and a as absorbing barriers.
Our goal is to compute the distribution of the final fortune. The key idea is that after each trial, the random process simply starts over again, but with a different initial fortune. This is an example of the Markov property and is fundamentally important in probability theory. Our analysis based on the Markov property suggests that we treat the initial fortune as a variable. Thus, we will denote the probability that the gambler reaches the target a, starting with an initial fortune x by
f(x) = P(XN = a | X0 = x) for x = 0, 1, ..., a.
1. By conditioning on the outcome of
the first trial, show that f satisfies the difference
equation
f(x) = qf(x - 1) + pf(x + 1) for x = 1, 2, ..., a - 1
and show that f satisfies the boundary conditions f(0) = 0, f(a) = 1.
The difference equation in Exercise 1 is linear, homogeneous and second order.
2. Show that the characteristic
equation of the difference equation in Exercise 1 is
pr2 - r + q = 0
and that the roots are r = 1 and r = q / p.
3. Show that if p is not 1/2
then the roots in Exercise 2 are distinct. Show that, in this
case, the probability that the gambler reaches his target is
4. Show that if p = 1/2, the
characteristic equation has a single root 1 that has multiplicity
2. Show that, in this case, the probability that the gambler
reaches his target is simply the ratio of the initial fortune to
the target fortune:
f(x) = x / a for x = 0, 1, ..., a.
From Exercises 3 and 4, we have the distribution of X in all cases:
P(XN = 0 | X0 = x) = 1 - f(x), P(XN = a | X0 = x) = f(x)
5. In the red and black experiment, choose Timid
Play and set a = 32 and p = 0.45. Vary x
form 0 to 32 with the scroll bar and note how the distribution of
the final fortune changes. Now with x = 24, run the
experiment 1000 times with an update frequency of 100 and note
the apparent convergence of the relative
frequency function to the true density.
6. Show that as a function of x,
for fixed p and a, f(x) increases
from 0 to 1 as x increases from 0 to a.
7. In the red and black experiment,
choose Timid Play and set a = 64 and x =
16. Vary p from 0 to 1 with the scroll bar and note how
the distribution of the final fortune changes. Now with p
= 0.52, run the experiment 1000 times with an update frequency of
100 and note the apparent convergence of the relative frequency
function to the true density.
8. Show that f(x) is
continuous as a function of p, for fixed x and a.
Specifically, use L'Hospital's Rule to show that the expression
in Exercise 3 converges to the expression in Exercise 4, as p
converges to 1/2.
9. In the red and black experiment, choose Timid
Play and set a = 64 and x = 32. Vary p
from 0 to 1 with the scroll bar and note how the distribution of
the final fortune changes. Now with p = 0.49, run the
experiment 1000 times with an update frequency of 100 and note
the apparent convergence of the relative frequency function to
the true density.
10. Show that for fixed x and a,
f(x) increases from 0 to 1 as p increases
form 0 to 1.
What happens if the gambler makes constant bets, but with an amount higher than $1? The answer to this question may give insight into what will happen with bold play.
11. In the red and black game, set the target
fortune to 16, the initial fortune to 8, and the win probability
to 0.45. Play 10 games with each of the following strategies.
Which seems to work best?
We will need to embellish our notation to indicate the dependence on the target fortune:
f(x; a) = P(XN = a | X0 = x)
Now fix p and suppose that the target fortune is 2a and the initial fortune is 2x. If the gambler plays timidly, then of course, his probability of reaching the target is f(2x; 2a). On the other hand:
12. Suppose that the gambler bets $2
on each trial. Argue that
Xi / 2, i = 0, 1, 2, ...
corresponds to timid play with initial fortune x and target fortune a and that therefore the probability that the gambler reaches the target is f(x; a)
Thus, we need to compare the probabilities f(2x; 2a) and f(x; a).
13. Show that
and show that
f(2x; 2a) < f(x; a) if p < 1 / 2; f(2x; 2a) > f(x; a) if p > 1 / 2.
Thus, it appears that increasing the bets is a good idea if the trials are unfair, a bad idea if the trials are favorable, and makes no difference if the trials are fair.
14. Generalize Exercises 12 and 13 to
compare timid play with the strategy of betting $k on each
trial (let the initial fortune be kx and the target
fortune ka).
Red and Black |
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