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I'm wondering how to go about solving this problem.

Flip a coin five times. Let $X_i$ be the indicator variable ($X_i = 1$) if the $i$th flip and $i+1$ flips are the same where $1\leq i \leq 4$. What is $E[X_i]$?

I'm thinking $X$ is 4 which is the number of $i$ and $i+1$ pairs so we have $X = X_1 + X_2 +...+ X_4$ but I don't know where to go from there or if that's the right setup.

yololo
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  • Welcome to MSE! What have you tried? Do you have any ideas of your own? Once we have a better idea of exactly where you're struggling, we can help you better ^_^ – HallaSurvivor Apr 24 '21 at 05:27

1 Answers1

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$X_i$ is the indicator that the event "flips $i$ and $i+1$ are the same" occurs.

Thus its expectation is the probability that this event occurs.

Let $H_i$ be the event that flip $i$ is a head. Then we have:

$$\forall i\in\{1,2,3,4\}~:~\mathsf E(X_i)=\mathsf P\big((H_i\cap H_{i+1})\cup(H_i^\complement\cap H_{i+1}^\complement)\big)$$

Graham Kemp
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