Multiplication Rule of Probability

Example 2

A graduate student in Statistics wanted to see if any correlation existed between a person's mathematical ability and their interest in the sciences. The following table shows the results of the study involving 200 student records, including the totals.

Math	Interest in the Sciences
Ability	Low	Average	High	Total
Low	52	10	16	78
Average	20	22	24	66
High	8	14	34	56
Total	80	46	74	200

a.   What is the probability of selecting a student with a high ability
in mathematics?

b.   What is the probability of selecting a student with a high ability
in math and a low interest in the sciences?

c.   What is the probability that a student with an average interest
in the sciences has an average ability in math?

d.   Does having a low ability in mathematics affect a student's likelihood
of having a low interest in the sciences?

 

 

 

 

 

 

Analysis of the Problem

The sample space of this experiment is based upon the set of 200 (sample size = n = 200) student records. The "experiment" consists of categorizing each student into his/her ability in mathematics and interest in the sciences.

The questions ask for probabilities. In order to obtain any of the marginal or joint probabilities from the table, we must divide all entries in the table by 200, which is the sample size. This gives us the following.

Math	Interest in the Sciences
Ability	Low (LS)	Average (AS)	High (HS)	Total
Low (LM)	0.26	0.05	0.08	0.39
Average (AM)	0.10	0.11	0.12	0.33
High (HM)	0.04	0.07	0.17	0.28
Total	0.40	0.23	0.37	1.00

The probability of selecting a student with a high ability in mathematics is 0.28. NOTE THAT this is a marginal probability.

The probability of selecting a student with a high ability in math and a low interest in the sciences is 0.04. NOTE THAT this is a joint probability.

Given that a student has an average interest in the sciences, the probability that he/she has an average ability in math is a conditional probability which may be derived as follows.

The probability that a student with an average interest in the sciences has an average ability in math is 0.478.

Does having a low ability in mathematics (event L_M) affect a student's likelihood of having a low interest in the sciences (event L_S) is simply another way of asking whether the two events L_M and L_S are independent or not.

We deduce that the two events are NOT independent.

Having a low ability in mathematics does affect a student's likelihood of having a low interest in the sciences, according to these particular statistics.

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Example 1

 

Last Updated by SAS October 07, 2026, 16:00:54