A hat contains slips of paper with the names of the 26 other students in Eduardo's class on them, 10 of whom are boys. To determine his partners for the group project, Eduardo has to pull two names out of the hat without replacing them. What is the probability that both of Eduardo's partners for the group project will not be boys?
Answer (2)
Determine the number of girls: 26 − 10 = 16 .
Calculate the probability of picking a girl first: 26 16 .
Calculate the probability of picking a girl second, given a girl was picked first: 25 15 .
Multiply the probabilities: 26 16 ⋅ 25 15 = 65 24 .
Explanation
Understand the problem and provided data Let's analyze the problem. Eduardo needs to pick two partners from his class of 26 students. We want to find the probability that both of the partners he picks are not boys. We know that there are 10 boys in the class, which means there are 26 - 10 = 16 girls.
Outline the solution To solve this, we need to find the probability of picking a girl first, and then picking another girl from the remaining students.
Calculate the probability of picking a girl first The probability of picking a girl on the first pick is the number of girls divided by the total number of students: P ( G 1 ) = 26 16
Calculate the probability of picking a girl second After picking one girl, there are now 15 girls left and a total of 25 students remaining. So, the probability of picking another girl on the second pick, given that the first pick was a girl, is: P ( G 2 ∣ G 1 ) = 25 15
Calculate the overall probability To find the probability that both picks are girls, we multiply the probabilities: P ( G 1 and G 2 ) = P ( G 1 ) ⋅ P ( G 2 ∣ G 1 ) = 26 16 ⋅ 25 15
Simplify the fraction and find the final probability Now, let's simplify the fraction: 26 16 ⋅ 25 15 = 13 8 ⋅ 5 3 = 65 24 So, the probability that both of Eduardo's partners are not boys is 65 24 .
Examples
Imagine you're organizing teams for a project and want to ensure diversity. If you know the proportion of different groups in a class, you can calculate the probability of forming a team with members from specific groups. This helps in creating balanced and inclusive teams.
Eduardo needs to pick two partners from his class and we calculated the probability that both are girls. With 16 girls in a class of 26, the probability that both picks are girls is \frac{24}{65}. This answer is calculated by finding the probability of picking girls consecutively without replacement.
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