Question

IF P(A)=1/3, P(B)=2/5, and P(AuB)=3/5, what is P(A^B)?

Answer 1

The probability of the union of two events is the sum of their probability, minus the probability of their interserction:

$P(A \cup B) = P(A) + P(B) - P(A \cap B)$

If we plug the known values into this formula, we have

$\dfrac{3}{5} = \dfrac{1}{3} + \dfrac{2}{5} - P(A\cap B)$

From which we can deduce

$P(A\cap B) = \dfrac{1}{3} + \dfrac{2}{5} - \dfrac{3}{5} = \dfrac{2}{15}$

So, the probability of $A \setminus B$ is a bit less than $P(A)$, we have to take away all events that belong to B as well:

$P(A \setminus B) = P(A) - P(A\cap B) = \dfrac{1}{3} - \dfrac{2}{15} = \dfrac{1}{5}$