Question

2. Given P(E) = 0.25, P(F) = 0.6, and P(E ∪ F) = 0.7.

Find:

a. What is P(E ∩ F)?

b. Are event E and event F mutually exclusive? Justify your answer.

c. Are event E and event F independent? Justify your answer.

Answer 1

$P(E \cup F) = P(E) + P(F) - P(E \cap F)$. From here we have:

a. $P(E \cap F) = P(E) + P(F) - P(E \cup F) = 0.25+0.6-0.7 =0.15$

b. If events are mutually exclusive, then $P(E \cap F) =0$. As we see from (a), $P(E \cap F) \neq 0$, therefore these events are not mutually exclusive.

c. If events are independent, then $P(E \cap F) = P(E) \cdot P(F)$. Let's check this.

$P(E) \cdot P(F) = 0.25 \cdot0.6 = 0.15$ and $P(E \cap F)= 0.15$.

Indeed, $P(E \cap F) = P(E) \cdot P(F)$, so these events are independent.