Question

Suppose that it is known that:

P(A) = 0.0261
P(B | A) = 0.375
P(B | ¬A) = 0.073
What is the probability of P(A | B)? State the answer.

Answer 1

Using Bayes' theorem and the given probabilities, P(A | B) is calculated by first finding P(B) with the law of total probability, then substituting into the formula P(A | B) = (P(B | A) × P(A)) / P(B).

Step-by-step explanation:

The student is asking for the probability of event A occurring given that event B has occurred, which is represented as P(A | B). To find this, we need to apply Bayes' theorem. The formula for Bayes' theorem is:

P(A | B) = (P(B | A) × P(A)) / P(B)

However, we are given P(B | A), P(B | ¬A), and P(A), but not P(B). We can find P(B) using the law of total probability:

P(B) = P(B | A) × P(A) + P(B | ¬A) × P(¬A)

Since P(¬A) is simply 1 - P(A), we can substitute the given values:

P(B) = 0.375 × 0.0261 + 0.073 × (1 - 0.0261)

Then, compute P(A | B) using the values for P(B | A), P(A), and the calculated P(B).