Question Video: Finding Conditional Probabilities Involving Two-Way Frequency Tables Mathematics • 10th Grade

Daniel and Jennifer are running for the presidency of the Students’ Union at their school. The votes they received from each of 3 classes are shown below. What is the probability that a student voted for Jennifer given that they are in the class B?

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Video Transcript

Daniel and Jennifer are running for the presidency of the Students’ Union at their school. The votes they received from each of three classes are shown in the table. What is the probability that a student voted for Jennifer given that they are in the class B?

Remember, the phrase “given that” indicates that we’re working with conditional probability. We’ll say that event A is that a student voted for Jennifer. And event B is that the student is in class B. Then this vertical line means given that. We’re finding the probability that A occurs given that B has already occurred. And one way we can do this is to narrow down the table based on the information that we’ve been given.

We’re told that that student is in class B, so we narrow it down to everyone in class B. In this case, we’re interested in the number of students that voted for Jennifer. That’s 195. And remember, to find the probability of an event occurring, we divide the number of ways that event can occur by the total number of outcomes. Here, the total number of possible outcomes is the total number of students in class B. That’s 169 plus 195, which is 364. And so the probability that a student voted for Jennifer given that they’re in class B is 195 over 364, which simplifies to 15 over 28.

Now, in fact, this is a perfectly valid method for answering this question. But there is a formula we can use. We say that to find the probability of A given B, we divide the probability of A intersection B ⁠— in other words, A and B ⁠— by the probability of B. So in this case, what’s A intersection B? Well, A was the number of students who voted for Jennifer and B was the number of students in class B. We’re looking for the intersection, the students that voted for Jennifer and are in class B. There are 195 of them.

The probability of choosing one of these at random is found by dividing 195 by the total number of students asked altogether. That’s 507 plus 494. That gives us a total of 1001. So the probability of A intersection B, in other words, the probability that a student voted for Jennifer and are in class B, is 195 out of 1001. And what about the probability of B, in other words, the probability that they’re in class B? Well, we already saw that there are 364 students in class B. So the probability of B occurring is 364 divided by 1001. And so, if we apply the formula, we get 195 over 1001 divided by 364 over 1001. Notice that this gives us the exact same answer as earlier, 195 over 364, which simplifies to 15 over 28.

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