Question

There are 6 girls and 7 boys in a class. A team of 10 players is to be selected from the class. If the selection is random, what is the probability of selecting a team of 5 girls and 5 boys?

Answer 1

The probability of picking 5 girls and 5 boys is `63/143`.

Explanation:

We cannot use the binomial distribution for this problem because the team members are picked WITHOUT replacement.

If we picked 5 girls and 5 boys, we would be left with 1 girl and 2 boys unpicked. Perhaps it is simpler to think of this problem this way - if we pick at random, what is the probability of having 1 girl and 2 boys unpicked. We can simply find the probability for picking the unpicked boys and girls. There are only 3 ways to select 1 girl (G) and 2 boys (B).

First, pick a girl. We have 6 girls and 13 total people, so the odds of picking a girl are 6/13. Next pick a boy. There would be 7 boys and only 12 people left, so this probability would be 7/12. Now pick another boy - 6 boys out of 11 people gives the probability of 6/11. So the probability for this case is

P(GBB) = `6/13*7/12*6/11`.

Similarly we could pick a boy first and then a girl and then another boy. The probability for this is

P(BGB) = `7/13*6/12*6/11`.

and for the last case,

P(BBG) = `7/13*6/12*6/11`.

So the total probability is

P(GBB) + P(BGB) + P(BBG) = `(3*7*6*6)/(13*12*11)=63/143`.