What Is the Addition Rule for Probabilities?

The addition rule for probabilities describes two formulas, one for the probability for either of two mutually exclusive events happening and the other for the probability of two non-mutually events happening. The first formula is just the sum of the probabilities of the two events. The second formula is the sum of the probabilities of the two events minus the probability that both will occur.

The Formulas for the Addition Rules for Probabilities Is

Mathematically, the probability of two mutually exclusive events is denoted by:

P(Y or Z)=P(Y)+P(Z)P(Y \text{ or } Z) = P(Y)+P(Z)P(Y or Z)=P(Y)+P(Z)

Mathematically, the probability of two non-mutually exclusive events is denoted by:

P(Y or Z)=P(Y)+P(Z)−P(Y and Z)P(Y \text{ or } Z) = P(Y) + P(Z) - P(Y \text{ and } Z)P(Y or Z)=P(Y)+P(Z)−P(Y and Z)

What Does the Addition Rule for Probabilities Tell You?

To illustrate the first rule in the addition rule for probabilities, consider a die with six sides and the chances of rolling either a 3 or a 6. Since the chances of rolling a 3 are 1 in 6 and the chances of rolling a 6 are also 1 in 6, the chance of rolling either a 3 or a 6 is:

  • 1/6 + 1/6 = 2/6 = 1/3

To illustrate the second rule, consider a class in which there are 9 boys and 11 girls. At the end of the term, 5 girls and 4 boys receive a grade of B. If a student is selected by chance, what are the odds that the student will be either a girl or a B student? Since the chances of selecting a girl are 11 in 20, the chances of selecting a B student are 9 in 20 and the chances of selecting a girl who is a B student are 5/20, the chances of picking a girl or a B student are:

  • 11/20 + 9/20 - 5/20 =15/20 = 3/4

In reality, the two rules simplify to just one rule, the second one. That's because in the first case, the probability of two mutually exclusive events both happening is 0. In the example with the die, it's impossible to roll both a 3 and a 6 on one roll of a single die. So the two events are mutually exclusive.

Key Takeaways

  • The addition rule for probabilities consists of two rules or formulas, with one that accommodates two mutually-exclusive events and another that accommodates two non-mutually exclusive events.
  • Non-mutually-exclusive means that some overlap exists between the two events in question, and the formula compensates for this by subtracting the probability of the overlap, P(Y and Z), from the sum of the probabilities of Y and Z.