# How to use Wald's equation to determine expectation in gambling model?

$$\begin{array}{l}{\text { In each game played one is equally likely to either win or lose 1. Let } S \text { be your }} \\ {\text { cumulative winnings if you use the strategy that quits playing if you win the first }} \\ {\text { game, and plays two more games and then quits if you lose the first game. }} \\ {\text { (a) Use Wald's equation to determine } E[S] \text { . }} \end{array}$$

Let $$X_i$$ be the amount won in game $$i$$. $$E[S] =E\bigg[\sum_{i=1}^N X_i\bigg]$$ Applying Walds theorem, $$=E[N]E[X]$$ And since this is a fair game where the player starts with 0 dollars, we know that $$E[X]=0$$. So, $$E[S]=0$$

Is this correct?

• Suppose $E[S]\neq0$, then this strategy (or its reverse) has a positive expected value on a fair game which is impossible So, in short, yes, it is true. – Stan Tendijck Apr 22 at 23:40
• @StanTendijck Thank you for verifying. Could you also take a look and verify the following if you are able? link to question – Jac Frall Apr 23 at 0:08