A group of eight grade 11 and five grade 13 students wish to be on the senior prom committee. The committee will consist of three students. What is the probability that only grade 12 students will be elected, assuming that all students have an equal chance of being elected?
There are a total of $8 + 5 = 13$ students from which to choose the committee members. Since only grade 12 students can be elected, there are 5 grade 12 students from which to choose the committee members. The probability that only grade 12 students will be elected is $\dfrac{\binom{5}{3}}{\binom{13}{3}} = \boxed{\dfrac{10}{143}}$.
To find the probability that only grade 12 students will be elected to the senior prom committee, we need to determine the total number of possible committees and the number of committees that consist solely of grade 12 students.
First, let's calculate the total number of possible committees. Since the committee will consist of three students, we can use the combination formula.
The total number of possible committees can be calculated as:
C(total students, committee size) = C(8+5, 3) = C(13, 3) = 13! / (3! * (13-3)!) = 13! / (3! * 10!) = (13 * 12 * 11) / (3 * 2) = 286
Now, let's calculate the number of committees that consist solely of grade 12 students. We have 8 grade 11 students and 5 grade 13 students, but all we need is 3 grade 12 students. Therefore, we can calculate the combination of selecting 3 grade 12 students from the available 8 grade 12 students (C(8, 3)).
The number of committees that consist solely of grade 12 students can be calculated as:
C(grade 12 students, committee size) = C(8, 3) = 8! / (3! * (8-3)!) = 8! / (3! * 5!) = (8 * 7 * 6) / (3 * 2) = 56
Now, we can calculate the probability by dividing the number of committees that consist solely of grade 12 students by the total number of possible committees:
Probability = number of desired outcomes / total number of possible outcomes
= number of committees that consist solely of grade 12 students / total number of possible committees
= 56 / 286
= 0.195
Therefore, the probability that only grade 12 students will be elected to the senior prom committee is approximately 0.195 or 19.5%.