probability

Question

1. Charlie Brown has 14 socks in his drawer, 7 blue & 6 green. He selects 5 socks at random. What is the probability that
   1. he gets two blue & three green?
   2. he gets three blue & two green?
   3. he gets two blue & three green, or 3 blue & 2 green?

Answer 1

Charlie has one odd sock which is neither blue nor green, because 14-(7+6)=1.

1. Let's suppose that he picks BBGGG in that order: sock 1: 7/14; sock 2: 6/13; sock 3: 6/12; sock 4: 5/11; sock 5: 4/10. Combined probability is the product of the individual probabilities: 3/143. But the same combination of socks can be made from the 10 permutations: BBGGG, BGBGG, GBBGG, GBGBG, GGGBB, GGBGB, GGBBG, BGGGB, BGGBG, GBGGB. So the actual probability of getting the same combination is 30/143, or 0.2098, 20.98%.

2. The number of permutations for 3 blue and 2 green as the same as (1). Socks 1 and 2 are the same as before: 7/14 and 6/13; sock 3: 5/12; sock 4: 6/11; sock 5: 5/10. Combined probability: 15/572. Combining permutations 10*15/572=75/286=0.2622, 26.22%.

3. To get (1) or (2) we add the probabilities: 135/286=0.4720, 47.20%.