riddle of the day

A game of chance can look like a tangle of possible paths. But if the game is fair, there is a way to see it without counting paths: your current money already contains the hidden probability of reaching the goal.

Fair Ruin

Strategist
Chance and uncertainty

Riddle statement

You have €7.

You play heads or tails against the casino.

In each round:

if you win, you gain €1;
if you lose, you lose €1.

The coin is fair.

You stop playing when one of two things happens:

you reach €10;
you go broke and reach €0.

What is the probability of reaching €10 before going broke?

Show solution

Solution

The probability is $7/10$, that is, 70%.

Let $p_k$ be the probability of reaching €10 before €0 when starting with $k$ euros.

At the endpoints:

$ p_0=0,\qquad p_{10}=1. $

For any intermediate amount, the next round moves with equal probability to the two neighboring states:

$ p_k=\frac{p_{k-1}+p_{k+1}}{2}. $

Therefore, each value in the sequence is the average of its neighbors. The only sequence that goes from 0 to 1 in ten steps with that property is linear:

$ p_k=\frac{k}{10}. $

Starting with €7:

$ p_7=\frac{7}{10}. $

Answer: the probability of reaching €10 before going broke is $7/10$.