Probability and combinations

Question

Aliens landed on Costco to buy some groceries and donate blood. The American president is sending an 8-member committee team of scientists to investigate this. His choices include 7 Biologists, 10 Chemists, 3 Physicists, and 4 Linguists. What is the probability that the team he selects consist:

a) All Biologists?

b) At least 2 Linguists?

c) Three times as many Chemists as there are Biologists?

Answer 1

7 + 10 + 3 + 4 = 24

In solving this problem, for the first two parts, you can either use combinatorics or the hypergeometric distribution. For the third, since we will have three different groups, combinatorics makes more sense.

a) Using combinatorics, you are selecting 7 of 7 biologists, and 1 of the 24 - 7 = 17 remaining.

P(all biologists) = C(7,7)C(17,1)/C(24,8) = 1 * 17/735471 = 1/43263 = 2.3114 * 10^-5; Excel uses combin for combination.

Alternatively, we could use = HYPGEOM.DIST(7,7,8,24,0) = 2.3114 * 10^-5 The final 0 means you are using the probability density function (often called pmf for discrete distributions) rather than cumulative probability.

b) Using combinatorics, we would probably do 1 - P(0 or 1). As there are 4 linguists, there are 24-4 = 20 others.

Then, 1 - (C(4,0)C(20,8) + C(4,1)C(20,7))/C(24,8) = 1 - (1*125970 + 4 * 77520)/735471 =

1 - 436050/735471 = 299421/735471 = 103/253 = .4071

Alternatively, as we are saying that, at most, 6 of the 20 others are selected,

=hypgeom.dist(6,20,8,24,1) = .4071

c) If there are 3 times as many chemists as biologists, and 8 total, we can have 0 of both, 4 of both, or 8 of both. There are 7 + 10 = 17 biologists and chemists, so 24 - 7 others

If we had 0 chemists and biologists, we would have to have 8 others, but there are only 7, so we can eliminate this possibility. I there are 4 chemists and biologists, there are fours others, and if there are 8, there are 0 other.

Then (C(10,3)C(7,1)C(7,4) + C(10,6)C(7,2)C(7,0))/C(24,8) = (120*7*35 + 210 * 21 * 1)/735471 =

33810/735471 = 490/10659 = .0460

Answer 2

If you look through the questions asked/answered last night, you'll see that this question was already answered then