Question

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

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

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

( p ( a ) = 0.25, p ( a ^ { \prime } ) = 0.75, p ( b | a ) = 0.5 ), and ( p ( b | a ^ { \prime } ) = 0.7 )

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

Explanation:

Step1: Substitute the values into the formula

We are given \(P(A) = 0.25\), \(P(A')=0.75\), \(P(B|A) = 0.5\), and \(P(B|A')=0.7\). Substitute these into the Bayes' Theorem formula \(P(A|B)=\frac{P(A)\cdot P(B|A)}{P(A)\cdot P(B|A)+P(A')\cdot P(B|A')}\).
So we get \(P(A|B)=\frac{0.25\times0.5}{0.25\times0.5 + 0.75\times0.7}\).

Step2: Calculate the numerator

Calculate \(0.25\times0.5=0.125\).

Step3: Calculate the denominator

First, calculate \(0.75\times0.7 = 0.525\). Then \(0.25\times0.5+0.75\times0.7=0.125 + 0.525=0.65\).

Step4: Calculate the final probability

Now, \(P(A|B)=\frac{0.125}{0.65}=\frac{125}{650}=\frac{5}{26}\approx0.192\).

Answer:

\(0.192\)