A class roster lists 15 boys and 12 girls. Two students are randomly selected to speak at a school assembly. if one of the students selected is boy, what is the probability that the other student selected is a girl?
6/13
12/(15-1+12) = ?
4/9
:3
To find the probability that the other student selected is a girl given that one of the students selected is a boy, we need to use conditional probability.
Let's break down the problem.
First, we are given that the class roster lists 15 boys and 12 girls. This information tells us the total number of students in the class, which is 15 boys + 12 girls = 27 students in total.
Next, we need to determine the sample space. The sample space represents all possible outcomes. In this case, since we are selecting two students to speak at the assembly, the sample space is the total number of ways we can choose two students from the class of 27. This can be calculated using the combination formula, which is denoted as:
nCr = n! / (r!(n - r)!)
where n is the total number of items to choose from, and r is the number of items being chosen.
Using this formula, we can calculate the total number of ways to choose two students from 27 as:
27C2 = 27! / (2!(27 - 2)!) = 27! / (2!25!) = (27 * 26) / (2 * 1) = 27 * 13 = 351
Therefore, the sample space is 351.
Now, let's consider the event where one of the selected students is a boy. Since we know one student is a boy, we need to calculate the probability that the other student selected is a girl. Let's call this event B (for the boy) and event G (for the girl).
The probability of event B can be calculated as the number of ways to choose one boy from 15 boys divided by the total number of ways to choose one student from the class:
P(B) = 15/27 = 5/9
Next, the probability of event G can be calculated as the number of ways to choose one girl from 12 girls divided by the remaining number of students after selecting the boy (which is 26):
P(G) = 12/26 = 6/13
Finally, the probability that the other selected student is a girl given that one of the students selected is a boy (P(G|B)) can be calculated using the formula for conditional probability:
P(G|B) = P(G ∩ B) / P(B)
where P(G ∩ B) represents the probability of both events G and B happening together.
Since these events are independent (the selection of one student does not affect the probability of selecting the other), we can multiply their probabilities:
P(G ∩ B) = P(G) * P(B) = (6/13) * (5/9) = 30/117
Therefore, the final probability is:
P(G|B) = (30/117) / (5/9) = 30/117 * 9/5 = 6/13
Therefore, the probability that the other student selected is a girl given that one of the students selected is a boy is 6/13.