Mr. Adams has 14 boys and 8 girls in his mathematics class. If he chooses two students at random to work on the board, what is the probability that both students chosen are boys? Show your work.

Answer 1

Heya Madi.

So the general way to calculate probabilities is to take the number of successes (the amount of outcomes that you want) and divide it by the number of total possible outcomes.

As a simple example, let's look at a coin. What is the probability of getting a heads with one coin flip?

Probability of heads

P(H) = # of heads / # of total outcomes

Well we know on a coin there is 1 head. And there are two possible outcomes, heads or tails. So 2 total outcomes.

P(H) = 1 / 2 = .5 = 50%

And the answer is 50%, which we know is indeed the chances of getting a heads on a coin flip.

So now let's take a look at our problem. We need to find the probability of choosing two boys.

Here, I will instead go through the process for finding the probability of choosing two girls. Then once you understand how it was done, hopefully you will be able to answer the actual question for yourself.

Finding the probability of choosing two girls. Let's make it simpler by starting with the probability of choosing the first girl only.

Probability of choosing the first girl. (Note: the letter chosen to represent the event is not important. Pick something that makes sense to you and be consistent)

P(G₁) = # of girls / # of total outcomes

So the number of girls we know to be 8. What is the number of total outcomes? Well a boy or a girl can be chosen, and there are 14 boys and 8 girls, for a total of 22 students.

P(G₁) = 8 / 22

Great. Now from here, let's find the probability of choosing the second girl.

P(G₂) = # of girls / # of total outcomes

However, this time things have changed. We've already chosen one girl, so the number of girls remaining to choose from is reduced 8 to 7. And the total number of students to choose from is reduced from 22 to 21.

P(G₂) = 7 / 21

Now we have the probability of choosing the first girl and the probability of choosing the second girl. From here, we just need to combine them to find the probability of both things happening together. To calculate the probability of two different events occurring, you multiply the two probabilities together.

Probability of choosing two girls. Let's use T for Two girls.

P(T) = P(G₁) × P(G₂) = (8 / 22) × (7 / 21)

From here you can use a calculator to find the answer. This answer can either be written as a fraction, a decimal, or a percentage.

Hope that was helpful. Let me know if you have any other questions.