Events and Probability Spaces · Recursive Probability Computation (Optional)
Lesson 5
Nikolai Chukhin · Alexander S. Kulikov
Let us apply our estimates to the casino game. Suppose the gambler starts with capital \(n=100\) and bets one coin on red each time, aiming to reach capital \(t=200\). What are their chances? The probability of winning a single bet in the casino is \(18/37\) or \(18/38\), i.e., slightly less than one-half (depending on whether one or two green sectors are added).
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Below, we will assume \(p=18/38\). Then \(r=20/18\). The probability of winning is then \[\frac{r^{100}-1}{r^{200}-1}=0.0000265607\dotsc\] This seems surprising: the probability of winning each individual bet is slightly less than one-half, yet the overall probability of winning is so small! In general, when \(p<\frac{1}{2}\), the probability of winning decreases exponentially with the increase in the intended profit \((t-n)\). Indeed, since \(r^{n}<r^{t}\) \[\Pr[\text{win}]=\frac{r^n-1}{r^t-1}<\frac{r^n}{r^t}=\frac{1}{r^{t-n}}.\] It is clear that this estimate depends only on the intended profit \(t-n\). For example, if the player wants to increase their capital by \(50\), their chances of doing so are no more than \[(20/18)^{-50}=0.005153775\dotsc,\] and this estimate holds for any initial capital \(n\)!For example, here is a plot of the probability of winning with initial capital \(n\) for a target capital \(t=200\). It is clear that the probability of winning is not close to zero only when the current capital \(n\) is close to the target.
import matplotlib.pyplot as plt
t, r = 200, 20/18
plt.plot([(r ** n - 1) / (r ** t - 1) for n in range(t + 1)])
plt.savefig('gambling_probability.png')
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