Question

According to Bayes' Theorem, the probability of event A, given that event B has occurred, is as follows. P(A∣B)= P(A)⋅P(B∣A)+P(A ′ )⋅P(B∣A ′ ) P(A)⋅P(B∣A) ​ Use Bayes' Theorem to find P(A∣B) using the probabilities shown below. P(A)= 3 2 ​ ,P(A ′ )= 3 1 ​ ,P(B∣A)= 7 1 ​ , and P(B∣A ′ )= 10 7 ​ The probability of event A, given that event B has occurred, is P(A∣B)= (Round to the nearest thousandth as needed.)

Answer 1

Using Bayes' Theorem and the given probabilities, the probability of event A given that event B has occurred is approximately 0.146 when rounded to the nearest thousandth.

Step-by-step explanation:

According to Bayes' Theorem, the probability of event A, given that event B has occurred, is calculated using the given probabilities. The correct formula for this calculation is:

P(A∣B) = \frac{P(A) × P(B∣A)}{P(A) × P(B∣A) + P(A') × P(B∣A')}.

Substituting the given probabilities into the formula:

P(A∣B) = \frac{\frac{2}{3} × \frac{1}{7}}{\frac{2}{3} × \frac{1}{7} + \frac{1}{3} × \frac{7}{10}} = \frac{\frac{2}{21}}{\frac{2}{21} + \frac{7}{30}} = \frac{\frac{2}{21}}{\frac{20}{63} + \frac{21}{63}} = \frac{\frac{2}{21}}{\frac{41}{63}} = \frac{2 × 63}{21 × 41} = \frac{2 × 3}{41} = \frac{6}{41}.

When converted to a decimal, this gives us:

P(A∣B) = 0.1463 (rounded to the nearest thousandth).

The conditional probability we are looking for, P(A∣B), is therefore approximately 0.146 when rounded to the nearest thousandth.