Andra M. answered • 11/26/22
Ivy League Tutor and mentor (Columbia BA, NYU PhD)
a) To find the probability that out of the 5 students we picked there is at least one girl, we will substract from 1 the probability that there are no girls (all boys).
P(5 students, at least 1 girl) = 1- P(all 5 boys)
We can write this out in 2 ways: first, by multiplying the probabilities as we pick each sequential boy:
P(all 5 boys) = 19/27 * 18/26 * 17/ 25 * 16/24 * 15/23 = (19!/ 14!) / (27!/ 22!)
Or, in the second way using the nchoose k formula :
P(all 5 boys) = P( 5 boys out of 19 boys)/ P(5 boys out of 27 students) = (19 choose 5)/ (27 choose 5) =
= [ 19! / (5!* 14!) ] / [ 27!/ (5!*22!)] = 11628/80730= 0.144
Notice that if we simplify by 5! in the above formula we get the same expression as in the first way.
Thus: P(5 students, at least 1 girl) = 1- 0.144 = 0.856
(b)
What are the ways in which we can get more boys than girls?
1): 5 boys
2) 4 boys, 1 girl
3) 3 boys, 2 girls
We will calculate all these probabilities and sum them up.
P(5 boys) = 0.144 as above.
P(4 boys, 1 girl) = [(19 choose 4) * (8 choose 1)] / (27 choose 5) = [ 19! / (4!*15!) * 8!/(1!*7!)] / [ 27!/ (5!*22!)]
We know from above that the term [ 27!/ (5!*22!)] = 80730. We will now simplify the top term of the fraction and compute it to get:
P(4 boys, 1 girl) = 31008/80730 = 0.384
P(3 boys, 2 girls ) = [(19 choose 3) * (8 choose 2)] / (27 choose 5) = [ 19! / (3!*16!) * 8!/(2!*6!)] / [ 27!/ (5!*22!)] =[ ( (17*18*19)/6 * (7*8)/2 ] / 80730 = 27132/80730 = 0.336
Summing up all these 3 probabilities:
P(more boys than girls) = P(5 boys) + P(4 boys, 1 girl) + P(3 boys, 2 girls ) = 0.144 + 0.384 + 0.336 = 0.864
This value makes sense, as out of the 27 students, there are 19 boys and only 8 girls, so when we choose 5 by chance it seems pretty likely (well, exactly with 0.864 probability) that we will choose more boys than girls.
