Question

the probability of event a, given that event b has occurred, can be found using bayess theorem.

p(a|b)=\frac{p(a)cdot p(b|a)}{p(a)cdot p(b|a)+p(a)cdot p(b|a)}

use bayess theorem to find ( p(a|b) ) using the probabilities shown below.

( p(a)=0.35, p(a) = 0.65, p(b|a)=0.5 ), and ( p(b|a) = 0.7 )

the probability of event a, given that event b has occurred, is (square).
(round to the nearest thousandth as needed.)

Explanation:

Step1: Substitute the values into the formula

We are given \(P(A) = 0.35\), \(P(A')=0.65\), \(P(B|A)=0.5\), and \(P(B|A') = 0.7\).
Substitute these into the formula \(P(A|B)=\frac{P(A)\cdot P(B|A)}{P(A)\cdot P(B|A)+P(A')\cdot P(B|A')}\)

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Step2: Calculate the numerator

Calculate \(0.35\times0.5=0.175\)

Step3: Calculate the denominator

First, calculate \(0.65\times0.7 = 0.455\)
Then, \(0.35\times0.5+0.65\times0.7=0.175 + 0.455=0.63\)

Step4: Calculate the final probability

$$P(A|B)=\frac{0.175}{0.63}\approx0.278$$

Answer:

\(0.278\)