Question

probability worksheet 4 (independent events, intersection rule)
1 multiple answer 1 point
if we have two events a and b, for them to be considered independent, which pair of probabilities need to be equal to each other? select all that apply
p(a) = p(a ∩ b)
p(b) = p(b | a)
p(a | b) = p(b | a)
p(a) = p(a | b)
p(a) = p(b)

Explanation:

For two events A and B to be independent, the occurrence of one event does not affect the probability of the other. The conditional - probability formula is \(P(A|B)=\frac{P(A\cap B)}{P(B)}\) and \(P(B|A)=\frac{P(A\cap B)}{P(A)}\). When A and B are independent, \(P(A\cap B)=P(A)\times P(B)\). So \(P(A | B)=\frac{P(A)\times P(B)}{P(B)} = P(A)\) and \(P(B | A)=\frac{P(A)\times P(B)}{P(A)} = P(B)\).

Answer:

B. \(P(B)=P(B|A)\)
D. \(P(A)=P(A|B)\)