Question

Let a and b be events in a sample space s such that p(a) = 23⁄100 , p(b) = 47⁄100 , and p(a ∩ b) = 3⁄20 . find p(b | ac ).

Answer 1

To find the probability of event B given the complement of event A (P(B | Ac)), one must first find P(Ac) and P(B ∩ Ac). After calculations, P(B | Ac) is determined to be 32/77.

Step-by-step explanation:

The question involves the concept of conditional probability in probability theory, a branch of mathematics. Given that P(A) = 23/100, P(B) = 47/100, and P(A ∩ B) = 3/20, we are asked to find P(B | Ac). The complement event Ac means that event A does not occur. Using the definition of conditional probability, P(B | Ac) = P(B ∩ Ac) / P(Ac).

First, we find P(Ac) which is the probability that event A does not happen,

P(Ac) = 1 - P(A) = 1 - (23/100) = 77/100.

Next, we need to find P(B ∩ Ac). We use the fact P(B) = P(B ∩ A) + P(B ∩ Ac),

Therefore, P(B ∩ Ac) = P(B) - P(B ∩ A) = (47/100) - (3/20).

Since 3/20 is equal to 15/100,

P(B ∩ Ac) = (47/100) - (15/100) = 32/100.

Finally, we calculate P(B | Ac),

P(B | Ac) = (32/100) / (77/100) = 32/77.

Therefore, the probability of event B occurring given that A does not occur is 32/77.