Question

Let (\Omega , F, P) be a probability space and let E₁ and E₂ 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 E₁ or E₂ will occur.

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

(c) None of E₁ and E₂ will occur

Answer 1

(a)

$$P(\text{Exactly one of the events }E_1\ or\ E_2)$$

$$=P(E_1)-P(E_1\cap E_2)+P(E_2)-P(E_1\cap E_2)$$

$$=0.2-0.1+0.4-0.1=0.4$$

(b)

$$P(\text{At least one of the events }E_1\ or\ E_2)$$

$$=P(E_1\cup E_2)=P(E_1)+P(E_2)-P(E_1\cap E_2)$$

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

(c)

$$P(\text{None of }E_1\ and\ E_2)$$

$$=1-P(E_1\cup E_2)=1-0.5=0.5$$