What is the difference between marginal and joint probability?
In Question 3 we introduced the idea of joint probability. Joint probabilities are the likelihoods associated with compound events using “and”. The joint probability of A and B is the likelihood that both events will occur simultaneously. Such probabilities are often encountered in consumer surveys. In this context, a consumer answers many questions about their behavior. In addition, the consumer also gives other information about themselves like gender, income, or educational attainment.
In several examples, we have looked at the results of a survey of 3743 consumers regarding data usage on cell phones. This survey also examined the gender of cell phone users with the results shown below.
Suppose we are interest in the joint probability that user is male and uses 2GB or more of data. The inclusion of the word “and” indicates we are interested in a compound event A and B where
A: the user is male
B: 2 GB or more of data is used
To find the joint probability , we need to calculate the relative frequency of the event A and B. Since there are 507 male users who use 2 GB or more of data and 3743 total users,
In this context, the probability of either of the individual events is called the marginal probability of the event. The marginal probability of A is calculated by finding the relative frequency of the event A.
Since there is a total of 2062 male users out of 3743 total users, the marginal probability is
The total number of male users is found by summing the shaded column values. In the example below, we use joint and marginal probabilities to find the likelihood of a compound event using “or”.
Example 6 Joint and Marginal Probability
In this example, we are interested in the events
C: the user is female
D: less than 200 MB of data is used
Use the information in the table below to calculate probabilities in each part.
a. Find the likelihood that a female user will use less than 200 MB of data.
Solution In terms of these events, we must find the probability that the user is female and less than 200 MB of data is used, P(C and D) . From the table, we recognize that there are 641 female users who use less than 200 MB of data. Since the total number of users is 3743, the relative frequency may be calculated,
The likelihood that a user in the survey is female and used less than 200 MB of data is approximately 17.1%.
b. Find the probability that a user is female.
Solution To use calculate the relative frequency of the event, we must divide the number of female users by the total number of users. The total number female users is at the bottom of the third column. Dividing this by the total number of users in the bottom of the last column gives,
The probability that a user in the survey is female is approximately 44.9%.
c. Find the probability that a user in the survey will use less than 200 MB of data.
Solution According to the survey, 1291 users of the total 3743 users used less than 200 MB of data. This means the probability of using less than 200 MB of data is
The likelihood of using less than 200 MB of data is approximately 34.5%.
d. Find the probability that the user is female or less than 200 MB of data is used.
Solution The event “the user is female or less than 200 MB of data is used” corresponds to the compound event C or D. We calculate the probability of this event by using the probabilities of the events in parts a through c,
The probability that a user is female or less than 200 MB of data is used is approximately 62.3%.