Multiplication Rule of Probability
Example 1
Forty-seven players entered a badminton tournament. In the table below, they are classified by gender and age.
Juniors |
Seniors |
|
Males |
15 | 11 |
Females |
10 | 11 |
a. What is the probability that a randomly selected player is a senior male?
b. What is the probability that a randomly selected player is a female?
c. What is the probability that a randomly selected female player is a junior?
Analysis of the Problem
The sample space of this experiment is based upon the set consisting of the entire group of badminton players. The "experiment" consists of categorizing each player of the group into his/her respective gender and age. Without belabouring the point, the contingency table gives the rest of the required information.
The total # of outcomes = sample size = n = 47.
The wording of the question asking for the probability of selecting a senior male means "senior and male". This should bring to mind joint probabilities and, coupled with it, the Multiplication Law. The contingency table given above should then be completed to include the "totals", or the marginal probabilities, as is shown below.
Juniors |
Seniors |
Total |
|
Males |
15 | 11 | 26 |
Females |
10 | 11 | 21 |
| Total | 25 | 22 | 47 |
Recalling the discussion on marginal probabilities, we can obtain the contingency table in terms of probabilities by dividing all entries in the table by the total number of players — namely 47 to obtain the following table. NOTE THAT S does not denote the sample space in this problem.
Juniors (event J) |
Seniors (event S) |
Total |
|
Males (event M) |
0.319 = P(M and J) | 0.234 = P(M and S) | 0.553 = P(M) |
Female (event F) |
0.213 = P(F and J) | 0.234 = P(F and S) | 0.447 = P(F) |
| Total | 0.532 = P(J) | 0.468 = P(S) | 1.00 |
Answer to 1.a
The answer comes immediately from the table. The probability that a randomly selected player is a senior male is 0.234. NOTE THAT this is a joint probability.
Answer to 1.b
The answer comes immediately from the table. The probability that a randomly selected player is a female is 0.447. NOTE THAT this is a marginal probability.
Solution to 1.c
The probability asked for in part c is a conditional probability. Reworded, the question may be phrased as: "Given that the selected player is a female, what is the probability that she is a junior player?"
It may be derived from the Multiplication Law as follows:
Answer to 1.c
The probability that a randomly selected female player is a junior 0.4765.
Last Updated by SAS October 07, 2026, 16:00:54
