Question 1150564: A history class is comprised of 5 female and 10 male students. If the instructor of the class randomly chooses 13 students from the class for an oral exam, what is the probability that 4 female students and 9 male students will be selected? Round your answer to 3 decimal places.
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website!
Answer: 0.476
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Explanation:
Because we dont care about ordering, we will use a combination instead of a permutation. The ranking of the individuals does not matter. The only thing we care about are the different groups. Within any single group, we can arrange the individuals however we want.
There are 5 female students, and we want to pick 4 of them. There are 5 ways to do this because nCr = 5 C 4 = 5.
This is equivalent to saying "there are 5 ways to not select any specific female student".
In general, nCr = n when r = 1 or when r = n-1. This leads to nice symmetry which is helpful.
The same idea applies to the males as well because 9 is one short of 10.
So nCr = 10 C 9 = 10; meaning there are 10 ways to pick 9 males from a pool of 10 total (equivalent to saying "there are 10 ways to not select a specific male student")
We have 5 ways to pick the four female students, and 10 ways to pick the nine male students, so there are 5*10 = 50 ways to pick the group of 4 female and 9 male students. We'll use this value 50 for later.
Overall, there are n = 5+10 = 15 students total. We want to pick a group of 13. So n = 15 and r = 13. Use the nCr combination formula.
n C r = (n!)/(r!*(n-r)!)
15 C 13 = (15!)/(13!*(15-13)!)
15 C 13 = (15!)/(13!*2!)
15 C 13 = (15*14*13!)/(13!*2!)
15 C 13 = (15*14)/(2!) .... notice how a pair of 13! terms cancel
15 C 13 = (15*14)/(2*1)
15 C 13 = 210/2
15 C 13 = 105
There are 105 ways to select the group of 13 students without any restrictions.
If we place restrictions (4 female, 9 male) on the group, then there are 50 different groups possible.
Divide the two values: 50/105 = 0.47619047619048
This rounds to 0.476
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