Modelling a fair game: A night at the Palais-Royal

Suppose that $M_{n}$ represents the amount of coins the player has at round $n$. For example, before they started playing, it was round “0”, and they had 100 coins in their wallet, so we’d write $M_{0} = 100$.

Then, they play the first round, and it lands on a heads! So the players gets a coin, and their wealth goes up by $1$, therefore $M_{1} = M_{0} + 1 = 101$. They play round 2, and it lands on a tails… they give a coin to the staff member, so their wealth goes down by $1$, so $M_{2} = M_{1} - 1 = 100$.

So the pattern emerges: in each round (let’s call it the $n$’th round), your current wealth ($M_{n}$) only changes by 1 from what you had the previous round ($M_{n-1}$), either going up, or going down.

Now, as the outcome on whether it goes up or down depends on a coin, the odds are 50/50 for heads or tails (we have assumed its a fair and unweighted coin).

Let’s now start thinking about what we might expect the player’s wallet to look like at any arbitrary round $n$. Well, we started at 100 coins at round 0, and , a coin is flipped, and it either went up ($+1$) or down ($-1$), and both outcomes are equally likely. So on average¹, if we were to play this game, starting from 100 coins, and going up to $n$ rounds, many times, we’d see a pattern that flips cancel each other out².

Let $\Omega = \{H,T\}$ and define our probability space as $(\Omega, \mathcal{P}(\Omega), \mathbb{P})$. Additionally, let $(X_{n})_{n \in \mathbb{N}}$ be a sequence of i.i.d random variables each defined $X_{n} : \Omega \to \{-1, 1\}$ such that $X_{n}(H) = 1$, and $X_{n}(T) = -1$.

Now, define the wealth of the player as

for $n \geq 1$, and set $M_{0} = 100$.

Note that we can just as easily express this wealth explicitly in terms of $(X_{n})_{n \in \mathbb{N}}$ and initial wealth $M_{0}$:

From this latter expression of the players wealth, it is immediately seen that the expected value of $M_{n}$ is constant, and equal to the initial wealth we started with, as

and as $\mathbb{E}(X_n) = 0$ for all $n$