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The probability Miss

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The probability Miss. Lula will run into the other room for no reason is 73 percent.
When Miss Lula's  sees her brush there is an 83 percent chance she will run to it and a 17 percent chance she will run into the other room. (cats are complicated creatures).
Miss Lula only sees the brush being taken out 68 percent of the time.
 
Given that Miss Lula has run into the other room what is the probability she saw her brush

pur-new-sol

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Answer:

Let the events be as follows:

A cat runs into a room = R

Miss Cat sees her toy = M

P(R|M') = 0.73

P(R|M) = 0.83

P(M') = 0.68

P(M|R') = P(R|M)*P(M)/[P(R|M)*P(M)+P(R|M')*P(M')]

= 0.83*0.32/[0.83*0.32+0.73*0.68]

= 0.2656/[0.2656+0.4964]

= 0.349

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