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Assume power failures occur independently of each other at a uniform rate through the months of the year, with little chance of $2$ or more occurring simultaneously. Suppose that $80\%$ of months have no power failures.

What is the probability that a month has more than one power failure.

I used a Poisson distribution with parameters $\lambda=0.2$ and $t=1$. Letting $X$ be the total number of failures in the month, I calculated $1-P(X=0)-P(X=1)$ and obtained $1-e^{-0.2}-0.2e^{-0.2}\approx 0.017523$. However, the book gives a solution of $0.0215$.

Which step was I wrong?

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up vote 3 down vote accepted

You incorrectly computed the value of $\lambda$. The equation you should have used $\Pr(X=0) = \exp(-\lambda) = 0.8$, giving $\lambda \approx 0.2231$.

Now $$ \Pr(X>1) = 1-\Pr(X=0)-\Pr(X=1) = 1 - 0.8 - 0.8 \lambda \approx 0.02148 $$

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