Question

Let A and B be two events such that

P(A) = 1/5 While P(A or B) = 1/2
Let P(B) = P. For what values of P
are A and B independent?​

Answer 1

We know that if A and B are independent, then the following equation holds true:
P(A and B) = P(A) * P(B)

Let's rewrite P(A or B) as follows:
P(A or B) = P(A) + P(B) - P(A and B)

We can plug in the given values to get:
1/2 = 1/5 + P - P(A and B)

We can simplify this equation to get:
3/10 = P - P(A and B)

Now, we need to find the values of P such that A and B are independent. This means that P(A and B) = P(A) * P(B). Substituting the given values, we get:
P(A) * P(B) = (1/5) * P

And from the above equation, we have:
P(A and B) = P - (3/10)

We can substitute these two equations and solve for P to get:
(1/5) * P = P - (3/10)

Solving for P, we get:
P = 3/4

Therefore, if P = 3/4, then A and B are independent events.