Video Transcript
Consider the two-way table showing how many men and women have pets and how many do not. Find the probability that a person chosen at random is a woman given that they have a pet.
The words “given that” in the question indicate that we are dealing with conditional probability. There are a couple of ways we could answer this question. We could do so using the formula for conditional probability. Or we could do it directly from the table. The two-way table is split into men and women, together with people that have pets and those who do not.
In this question, the condition is that the person must have a pet. And we are therefore only interested in the values in this column. We want to find the probability that the person chosen is a woman. As probability can be written as a fraction where the numerator is the number of successful outcomes and the denominator, the number of possible outcomes, then the probability in this question can be written as the number of women who have a pet over the number of people who have a pet.
We can see from the table that there are 78 women who have a pet. Since 22 plus 78 is equal to 100, there are 100 people that have a pet in total. The probability that a person chosen is a woman given that they have a pet is therefore equal to 78 out of 100. Dividing both the numerator and denominator by two, this simplifies to 39 over 50 or thirty-nine fiftieths.
As already mentioned, an alternative method to calculate this would be using the conditional probability formula. This states that the probability that 𝐴 occurs given that 𝐵 occurs is equal to the probability of 𝐴 intersection 𝐵 over the probability of 𝐵. In this question, we are trying to find the probability that the person is a woman given that they have a pet.
The intersection of these two events will be the women who have a pet. Once again from the table, we see that this is 78 out of the 200 people in total. We have also already calculated that there are 100 people that have a pet. So the probability of selecting a person who has a pet is 100 out of 200. Dividing 78 over 200 by 100 over 200 once again simplifies to 78 over 100. This confirms the answer that we have already calculated of 39 over 50. The probability that a person chosen at random is a woman given that they have a pet is thirty-nine fiftieths.
