SOLUTION: There are 7 men and 3 women in a room. Two of these 10 people are selected at random. If both people are the same gender, then what is the probability that they are both women?

Question 566612: There are 7 men and 3 women in a room. Two of these 10 people are selected at random. If both people are the same gender, then what is the probability that they are both women?
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
7 men and 3 women.
2 selected at random.
both are the same gender.
let p(a) be the probability that they are both women.
let p(b) be the probability that they are both the same gender.
the probability that they are both men is 7/10 * 6/9 = 42/90
the probability that they are both women is 3/10 * 2/9 = 6/90
the probability that they are the same gender is equal to 42/90 + 6/90 = 48/90.
the probability that they are the same sex and are both women at the same time is the same as the probability that they are both women.
the formula for conditional probability is:
p(a|b) = p(a^b)/p(b)
this formula means:
probability of a given b equals the probability of a and b occurring at the same time divided by the probability of b.
probability of a and b occurring at the same time is the probability that they are the same gender and are both women at the same time which is equal to 6/90.
the probability of b occurring is the probability that they are both of the same gender which is equal to 48/90.
p(a|b) = p(a^b)/p(b) becomes:
p(a|b) = (6/90) / (48/90) = (6/90) * (90/48) = 6/48 = .125

this might be easier to see in a smaller problem.
assume 5 people and 2 are picked at random.
3 are men and 2 are women.
probability that 2 people picked will be men is 3/5 * 2/4 = 6/20.
probability that 2 people picked will be women is 2/5 * 1/4 = 2/20.
probability that 2 people picked will be the same gender is equal to 6/20 + 2/20 = 8/20.
probability that 2 people picked will be the same gender and will be women at the same time is equal to 2/20 which is the same as the probability that 2 women picked will be both women.
probability that 2 people picked will be 2 women given that the people picked are the same gender is equal to probability they are the same gender and they are both women at the same time divided by the probability they are both of the same gender.
this becomes (2/20) / (8/20) which equals (2/30) * (20/8) which is equal to (2/8) which i equal to .25.
there is a .25 probability that, if the 2 people picked are the same gender, that they will be women.
with 5 people, it's easier to show you how it works rather than with 10.
here's the breakdown.
assume men are numbers 1, 2, and 3
assume women are letters a and b.
the possible 2 person combinations are:
1,2 ***
1,3 ***
1,a
1,b
2,3 ***
2,a
2,b
3,a
3,b
a,b ***
the persons of the same gender have an asterisk next to their selection.
there are 4 out of 10 possible selections that are the same gender.
out of these 4, 1 is both women.
the probability that, given they are the same gender, that they will both be women is equal to 1 pair of women out of 4 total possible pairs of same gender.
the same formula applies to 10 people where 7 are men and 3 are women as shown above.

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