Question

suppose that we have two events, a and b, with p(a)=0.50, p(b)=0.60, and p(a ∩ b)=0.40. 1. find p(a|b) 2. find p(b|a) 3. are a and b independent? why or why not?

Answer 1

The conditional probabilities are P(A|B) = 0.67 and P(B|A) = 0.80. Events A and B are not independent because P(A ∩ B) does not equal P(A)*P(B).

Step-by-step explanation:

Probability and Independent Events

To find P(A|B), which is the probability of event A given that event B has occurred, we use the formula: P(A|B) = P(A ∩ B) / P(B). Given that P(A) = 0.50, P(B) = 0.60, and P(A ∩ B) = 0.40:

P(A|B) = 0.40 / 0.60 = 2/3 or approximately 0.67.

To find P(B|A), which is the probability of event B given that event A has occurred, we use the formula similarly: P(B|A) = P(A ∩ B) / P(A).

P(B|A) = 0.40 / 0.50 = 4/5 or 0.80.

To determine if events A and B are independent, we check if the following is true: P(A ∩ B) = P(A)P(B). In our case:

P(A)P(B) = 0.50 * 0.60 = 0.30, which does not equal P(A ∩ B) = 0.40. Hence, events A and B are not independent, as the probability of their intersection does not equal the product of their individual probabilities.