conditional Probability

Question

The probability that event A occurs is 𝑃(𝐴) = 0.4. B is an event independent of A and

the probability of the union of A and B is 𝑃(𝐴 ∪ 𝐵) = 0.7. Find 𝑃(𝐴 ∩ 𝐵).?

Answer 1

For independent events A and B, we need the following formula:

𝑃(𝐴 ∪ 𝐵) = 𝑃(𝐴) + 𝑃(𝐵) - 𝑃(𝐴 ∩ 𝐵)

and

𝑃(𝐴 ∩ 𝐵) = 𝑃(𝐴) • 𝑃(𝐵)

Given:

𝑃(𝐴) = 0.4

𝑃(𝐴 ∪ 𝐵) = 0.7

Find: 𝑃(𝐴 ∩ 𝐵)

Using: 𝑃(𝐴 ∪ 𝐵) = 𝑃(𝐴) + 𝑃(𝐵) - 𝑃(𝐴 ∩ 𝐵)

0.7 = 0.4 + 𝑃(𝐵) - 𝑃(𝐴 ∩ 𝐵)

𝑃(𝐵) = 𝑃(𝐴 ∩ 𝐵) + 0.3

Using: 𝑃(𝐴 ∩ 𝐵) = 𝑃(𝐴) • 𝑃(𝐵)

𝑃(𝐴 ∩ 𝐵) = 0.7 • 𝑃(𝐵)

𝑃(𝐴 ∩ 𝐵) =(7/10) • 𝑃(𝐵)

(10/7)•𝑃(𝐴 ∩ 𝐵) = 𝑃(𝐵)

Put two equations together:

𝑃(𝐴 ∩ 𝐵) + 0.3 = (10/7)•𝑃(𝐴 ∩ 𝐵)

𝑃(𝐴 ∩ 𝐵) - (10/7)•𝑃(𝐴 ∩ 𝐵) = -0.3

-(3/7)•𝑃(𝐴 ∩ 𝐵) = -(3/10)

Multiply both sides by -7/3

𝑃(𝐴 ∩ 𝐵) = (7/3)(3/10) = 7/10 =0.7