Question

Let (2, F. P) be a probability space and let E, and Ey be two events with P(E₁) = 0.2. P(E₂) 0.4 and P(E₁ E₂)=0.1.

Find the probability that (a) Exactly one of the events E, or E₂ will occur.

(b) At least one of the events E, or E, will occur.

(c) None of E, and Es will occur.

Answer 1

Solution:

P(E₁) = 0.2. P(E₂) 0.4 and P(E₁$\cap$ E₂)=0.1.

(a) Exactly one of the events E, or E₂ will occur.

Required probability = P(E₁) + P(E₂) - 2P(E₁$\cap$ E₂)

$=0.2+0.4-2\times0.1 \\=0.6-0.2 \\=0.4$

(b) At least one of the events E₁ or E₂ will occur.

Required probability = P(E₁$\cup$ E₂) = P(E₁) + P(E₂) - P(E₁$\cap$ E₂)

$=0.2+0.4-0.1 \\=0.6-0.1 \\=0.5$

(c) None of E₁ and E₂ will occur.

Required probability = P(E₁' $\cap$E₂') = P[(E₁$\cup$ E₂)'] = 1-P[(E₁$\cup$ E₂)]

$=1-0.5 \\=0.5$