# What Is The Rule For Computing The Probability Of A Union Of Two Events?

Image of Whiskey Off-Road by Andrea Schaben

If E and F are events, we can compute the probability of the union of these events using

In an applied problem, you might see the word “or” used in place of the union ∪ symbol or the word “and” used in place of the intersection symbol ∩. In this case we can write out this fu=ormula as

P(E or F) = P(E) + P(F) – P(E and F)

If we know any three of the four probabilities in the formula, we can solve for the fourth probability. This formula can also be used if the probabilities involve compliments. In that case, we can utilize

to help compute the probability. The examples below illustrates how we can use these formulas.

Problem Your bicycle has a noise coming from the back wheel. The probability that the rear hub needs to be adjusted is 0.4. The probability that the brakes need adjusting is 0.55. The probability that the rear hub needs to be adjusted and the brakes need to be adjusted is 0.10.

a. What is the probability that the rear hub needs adjusting or the rear brakes need adjusting?

Solution Let’s start by defining the events in this problem.

R: the rear hub needs adjusting