Question 1162731: The first 5 students to arrive in a school on Monday morning were 2 boys and 3 girls, of these two were choose at random for an assignment. Find the probability that both are boys and the probability that the two were of different sex
Answer by jim_thompson5910(35256)   (Show Source):
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2 boys, 3 girls, 5 total
P(boy) = 2/5 is the probability of selecting a boy
P(girl) = 3/5, probability of selecting a girl
P(2 boys) = (2/5)*(1/4) = 2/20 = 1/10, probability of selecting 2 boys
P(2 girls) = (3/5)*(2/4) = 6/20 = 3/10, probability of selecting 2 girls
P(2 of same gender) = probability of selecting 2 of same gender
P(2 of same gender) = P(2 boys) + P(2 girls)
P(2 of same gender) = 1/10 + 3/10
P(2 of same gender) = 4/10
P(2 of same gender) = 2/5
The events "picking 2 boys" and "picking 2 girls" are mutually exclusive. One or the other can happen, but both cannot happen simultaneously. The general rule is that if A and B are mutually exclusive events, then P(A or B) = P(A)+P(B).
P(1 boy and 1 girl) = 1 - P(2 boys OR 2 girls)
P(1 boy and 1 girl) = 1 - 2/5
P(1 boy and 1 girl) = 5/5 - 2/5
P(1 boy and 1 girl) = 3/5
The order of selection does not matter.
Side note:
We have three scenarios
A) we get 2 boys
B) we get 2 girls
C) we get 1 of each gender
Scenarios A and B combine to get the "P(2 of same gender)" while scenario C is "P(1 boy and 1 girl)". Since scenarios A,B,C are the only possibilities, this means that the corresponding probabilities must add to 1. The events are complementary.
Alternatively, you can do it like this
P(first is a boy, second is a girl) = P(first is boy)*P(second is girl)
P(first is a boy, second is a girl) = (2/5)*(3/4)
P(first is a boy, second is a girl) = 3/10
P(first is a girl, second is a boy) = P(first is girl)*P(second is boy)
P(first is a girl, second is a boy) = (3/5)*(2/4)
P(first is a girl, second is a boy) = 3/10
Adding those two mutually exclusive events gives,
P(1 of each gender) = 3/10 + 3/10
P(1 of each gender) = 6/10
P(1 of each gender) = 3/5
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Answers:
Probability both are boys = 1/10
Probability selecting one of each gender = 3/5
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