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The probability that Docter Probs succeeds on a single surgery, which is independent from other surgeries, is $p$. Every time he performs 3 successful surgeries in a row, he shouts 'yay!'. Specifically, say if he performs 6 successful surgeries in a row, then he shouts 'yay' 4 times.

How many 'yay!'s is he expected to shout if he performs 100 surgeries in a row?

Here's what I've tried so far: The probability of any 3 surgeries in a row being successful is $p^3$. In 100 surgeries, he can have a maximum of 98 'yays!'. Hence, he will shout $98p^3$ 'yays!'.

However, I just realized that the probabilities of shouting yays are not independent. How do I take into account how one failed surgery could affect up to 3 yays?

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Let $Y_k$ be a random variable that takes the value $1$ if he shouts ‘yay!’ after the $k$-th surgery and $0$ if he does not. Clearly the expected value of $Y_k$ is $p^3$ if $3\le k\le 100$ and $0$ if $k$ is $1$ or $2$. Expectation is linear whether the variables are independent or not, so

$\Bbb E\left(\sum_{k=0}^{100}Y_k\right)=\sum_{k=1}^{100}\Bbb E(Y_k)=98p^3\;,$

just as you thought at first.

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You don't need to; this is the wonderful fact of linearity of expectation, even when you don't have independence.

http://en.wikipedia.org/wiki/Expected_value#Linearity See section 3.3.

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    Thanks, it makes sense in the context of expected value. However, it still doesn't make intuitive sense to me... aren't the probabilities dependent in this case?2019-02-24