Similar to the previous two posts, although the premise is slightly different. A gambler starts with n dollars and bets dollar 1 at a time. She will quit under one of the two circumstances – 1) lose it all (0 dollars) or 2) reaches the target of N dollars.
Let’s first understand the conditions. You go to a casino and play even money on a Roulette wheel (payoff 1 to 1). You have 10 dollars in your purse, which is your capital. You start betting 1 dollar at a time. If you win, you add a dollar to the capital, and if you lose the bet, you lose one from it. When you reach your target, say 100, or lose all your money, you quit and go home. It is easy to realise that you can not start if your starting capital is 0 or 100. In the former case, you don’t have money to bet, and in the latter, you have already achieved the target!
Random walk
We use the random walk method to establish an analytical relationship for the probability. Imagine a random walk starts from a position xj, corresponding to a starting fortune of j dollars. Depending on a win, loss, or a tie, the person will move to xj+1, xj-1 and xj with probabilities of p, q and r, respectively. Therefore,
xj = P(Aj|win) x P(win) + P(Aj|loss) x P(loss) + P(Aj|tie) x P(tie)
xj = xj+1 x p + xj-1 x q + xj x r
Total probability, p + q + r = 1; r = 1 – (p + q)
xj = xj+1 x p + xj-1 x q + xj – (p+q) xj
| p xj+1 – (p+q) xj + q xj-1 = 0 |
It is a quadratic equation for xj = k(j=1) . By substituting the values and performing the necessary manipulations, you get the final probability of reaching the target (quitting at N or 0).
For even-money bets
The winning probability is just under 50% (18/38). The chances of achieving your target of 100 from four different starting points, 10, 25, 50, and 93.5, are:
| Starting Amount | Probability (to reach 100) |
| 10 | 0.00005 |
| 25 | 0.0003 |
| 50 | 0.005 |
| 93.5 | 0.5 |
Bold vs cautious
An important takeaway from this calculation is the strategy of how you may want to bet to maximise your chance of reaching 100. E.g., you start with 10 dollars and have two choices: 1) place 10-dollar bets or 2) place 1-dollar bets. In the first case, you bet ten times, and in the second case, a hundred.
![\text{The probability of winning 100 in 10 dollar bets starting with 10 is (n = 1 and N = 10)} \\ \\ x_{10} = \frac{1-[(18/38)/(18/38)]^1}{1-[(18/38)/(18/38)]^{10}} = 0.06 \\ \\ \text{The probability of winning 100 in 1 dollar bets starting with 10 is (n = 10 and N = 100)} \\ \\ x_{10} = \frac{1-[(18/38)/(18/38)]^{10}}{1-[(18/38)/(18/38)]^{100}} = 0.00005](https://thoughtfulexaminations.com/wp-content/ql-cache/quicklatex.com-2f37e567272f922695016f9869199d9d_l3.png "Rendered by QuickLaTeX.com")
You better be bold and play larger sums fewer times than otherwise. Well, it is not new; the house always wins in the long term!
