Question

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

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

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

( p(a)=83%, p(a)=17%, p(b|a)=49% ), and ( p(b|a)=47% )

the probability of event a, given that event b has occurred, is
(round to three decimal places as needed.)

Explanation:

Step1: Substitute the values into the formula

Given \(P(A) = 0.83\), \(P(A')=0.17\), \(P(B|A)=0.49\), \(P(B|A') = 0.47\).
Substitute into \(P(A|B)=\frac{P(A)\cdot P(B|A)}{P(A)\cdot P(B|A)+P(A')\cdot P(B|A')}\)

$$
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Step2: Calculate the numerator

\(0.83\times0.49=0.4067\)

Step3: Calculate the denominator

First, \(0.17\times0.47 = 0.0799\)
Then, \(0.83\times0.49+0.17\times0.47=0.4067 + 0.0799=0.4866\)

Step4: Calculate the final probability

\(P(A|B)=\frac{0.4067}{0.4866}\approx0.836\)

Answer:

\(0.836\)