Question

the billing department of a major credit card company attempts to control errors on customers billa. Suppose that occur according to a poisson distribution with a parameter. what is the probability that a customers bill selected at random will contain one error?

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Formula Poisson probability\textbf{Poisson probability}:

P(X=k)=λkeλk!P(X=k)=\dfrac{\lambda^k e^{-\lambda}}{k!}

The errors occur according a Poisson distribution with some parameter λ\lambda.

The probability that a customers bill selected at random will contain one error is then the formula of Poisson probability evaluated at k=1k=1.

P(X=1)=λ1eλ1!=λeλ1=λeλP(X=1)=\dfrac{\lambda^1 e^{-\lambda}}{1!}=\dfrac{\lambda e^{-\lambda}}{1}=\lambda e^{-\lambda}

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