According to Bayes' Theorem, the probability of event A, given that event B has occurred, is as follows.

Upper P left parenthesis Upper A vertical line Upper B right parenthesis equals StartFraction Upper P left parenthesis Upper A right parenthesis times Upper P left parenthesis Upper B vertical line Upper A right parenthesis Over Upper P left parenthesis Upper A right parenthesis times Upper P left parenthesis Upper B vertical line Upper A right parenthesis plus Upper P left parenthesis Upper A prime right parenthesis times Upper P left parenthesis Upper B vertical line Upper A prime right parenthesis EndFractionP(AB)=P(A)P(BA)P(A)P(BA)+PAPBA

Use Bayes' Theorem to find

Upper P left parenthesis Upper A vertical line Upper B right parenthesisP(AB)

using the probabilities shown below.

Upper P left parenthesis Upper A right parenthesis equals five sixthsP(A)=56,

Upper P left parenthesis Upper A prime right parenthesis equals one sixthPA=16,

Upper P left parenthesis Upper B vertical line Upper A right parenthesis equals one tenthP(BA)=110,

and Upper P left parenthesis Upper B vertical line Upper A prime right parenthesis equals one halfPBA=12

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Part 1

The probability of event A, given that event B has occurred, is

Upper P left parenthesis Upper A vertical line Upper B right parenthesisP(AB)equals=enter your response here.

(Round to the nearest thousandth as needed.)