Question

If a and b are independent events, p(a) = 0.25, and p(b) = 0.65, find the following probabilities.

(a) p(a ∩ b)
(b) p(a ∪ b)
(c) p(a | b)
(d) p(Ac ∪ Bc)

Answer 1

For independent events a and b with given probabilities, the probability of their intersection is 0.1625, the probability of their union is 0.7375, the conditional probability of a given b is 0.25, and the probability of the complement of a union the complement of b is 0.5625.

Step-by-step explanation:

Solution:

(a) To find the probability of the intersection of two independent events, we multiply their individual probabilities. Therefore,

P(a ∩ b) = P(a) × P(b) = 0.25 × 0.65 = 0.1625.
(b) To find the probability of the union of two independent events, we subtract the probability of their intersection from the sum of their individual probabilities. Therefore,

P(a ∪ b) = P(a) + P(b) - P(a ∩ b) = 0.25 + 0.65 - 0.1625 = 0.7375.
(c) To find the conditional probability of event a given event b, we divide the probability of their intersection by the probability of event b. Therefore,

P(a | b) = P(a ∩ b) / P(b) = 0.1625 / 0.65 = 0.25.
(d) To find the probability of the complement of event A union the complement of event B, we subtract the probability of event A from 1 and subtract the probability of event B from 1, and then find the intersection of the complements. Therefore,

P(Ac ∪ Bc) = (1 - P(a)) × (1 - P(b)) = (1 - 0.25) × (1 - 0.65) = 0.5625.