Question Video: Using Combinations to Find the Probability of an Event

Video Transcript

13 boys and 15 girls entered a draw for five free movie tickets. What is the probability that all five tickets were won by boys?

This question is about the number of possible combinations of people who could win the five free movie tickets. In order to find the probability that all five tickets were won by boys, we need to calculate two things: firstly the total number of possible combinations for the five ticket winners regardless of whether they’re boys or girls, secondly the number of possible combinations where all the winners are boys.

Let’s recall first of all how to calculate a combination. The number of possible combinations in which 𝑟 objects are taken from 𝑛 is given by 𝑁𝐶𝑅, which is calculated as 𝑛 factorial divided by 𝑟 factorial and 𝑛 minus 𝑟 factorial.

Remember 𝑁𝐶𝑅 is normally read as 𝑛 choose 𝑟. So firstly, the total number of possible combinations for the five ticket winners. Here we’re not interested in whether the winners are girls or boys, so we just want the number of possible combinations of choosing five people from the total, which is 28. So the total number of possible combinations is 28 choose five. You can use your calculator to evaluate this; it’s 98280.

Next, let’s think about the number of possible combinations where all the winners are boys. This means that the five ticket winners need to be chosen from the 13 boys. Therefore, the number of possible combinations where all the winners are boys is 13 choose five, which is calculated as 13 factorial over five factorial and eight factorial. Again you can use your calculator to evaluate this; it gives 1287.

Now remember the question asked us for the probability that all five tickets were won by boys, which means we need to divide the number of possible combinations where all winners are boys by the total number of possible combinations.

So we have 1287 divided by 98280. This can be simplified. So as a fraction in its simplest form, the probability that all five tickets were won by boys is 11 over 840.