probability questions

Question

A city council consists of six Democrats and eight Republicans. If a committee of seven people is selected, find the probability of selecting four Democrats and three Republicans.

Answer 1

This question is a little tricky because we have to figure out how many groups of the total set contain the required number of each party.  Luckily the fact that picking a group of 4 democrats from the 6 is independent from picking 3 republicans from the eight makes this calculation easier.

 

The number of groups of seven with 4 democrats and 3 republicans = (number of groups of 4 democrats out of the 6 possible) × (number of groups of 3 republicans out of the eight possible)

Since the order we choose people for the group doesn't matter, this is a combination problem.

 

Democrat groups = ₆C₄ = 6!/[(6-4)!4!] = 6!/(2!4!) = 15

Republican groups = ₈C₃ = 8!/[(8-3)!3!] = 8!/(5!3!) = 56

 

Therefore the total number of groups that meet our requirement is 15*56 = 840

 

Now we need to figure out how many groups of 7 are possible, regardless of party, out of the 14 total.  This is just ₁₄C₇ = 14!/[(14-7)!7!) = 14!/(7!7!) = 3432 different groups.

 

The probability of getting a group with the desired makeup would be 840/3432 = 35/143 ≈ 0.245 or 24.5%