What is the addition rule for probabilities?
The addition rule for probabilities describes two formulas, one for the probability that one of the two mutually exclusive events will occur and the other for the probability that two non-mutually exclusive events will occur. The first formula is only 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 are
Mathematically, the probability of two mutually exclusive events is noted by:
P(Yes or Z)=P(Yes)+P(Z)The
Mathematically, the probability of two non-mutually exclusive events is noted by:
P(Yes or Z)=P(Yes)+P(Z)–P(Yes and Z)The
What does the addition of probabilities rule tell you?
To illustrate the first rule of the addition of probabilities rule, consider a six-sided die and the chances of rolling 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 getting 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 chances that the student will be a girl or a student B? Since the chances of selecting a girl are 11 out of 20, the chances of selecting a student B are 9 out of 20 and the chances of selecting a girl who is a student B are 5/20, the chances of choosing a girl or student B are:
- 11/20 + 9/20 – 5/20 = 15/20 = 3/4
In reality, the two rules are simplified into a single rule, the second. This is because in the first case, the probability that two mutually exclusive events will both occur is 0. In the example with the die, it is impossible to roll both a 3 and a 6 on a single roll of a die. The two events are therefore mutually exclusive.
Key points to remember
- The addition rule for probabilities is made up of two rules or formulas, one which supports two mutually exclusive events and another which supports two non-mutually exclusive events.
- Not mutually exclusive means that there is some overlap 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.