Let A, B and C be the three mutually independent events with their probabilities equal to x, y and z respectively. Then the probability that exactly two of these events occur is

Given the probabilities of three mutually independent events as x, y and z, respectively.

Probability of the first event happening = x

Probability of not happening of the first event = 1-x

Probability of the second event happening = y

Probability of not happening of the second event = 1-y

Probability of the third event happening = z

Probability of not happening of the third event = 1-z

Probability of exactly 2 events happening = $$\ \ \ \ \ xy\left(1-z\right)+yz\left(1-x\right)+zx\left(1-y\right)$$

= $$xy+yz+zx-3xyz$$