In a bag of 100 chips, three are black. If 6 chips are drawn from the bag, what is the chance that a black one will be drawn?

Answer 1

It is 1 - the probability of selecting no black chips. There are 3 black chips, and 100 chips altogether, so there are 97 non-black chips. If we select 6 chips, there are C(97,6) sets of non-black chips, and C(100,6) sets altogether, so the chance of selecting all non-black chips = C(97,6)/C(100,6) = P(97,6)/P(100,6) = P(97,3)P(94,3)/(P(100,3)P(97,3)) =

P(94,3)/P(100,3) = 94*93*92/(100*99*98) = 47*31*23/(49*33*25) = 33511/40425

Then, the chance of a black chip is 1 - 33511/40425 = 6914/40425 = .171033

Answer 2

C(n,x) is number of combinations of x out of n

n!

----

x! (x-n)!

want to select 1 out of the 3 black chips. C(3,1) is the number of combinations of 1 black chip out of 3.

want to select 5 out of the remaining 97 non-black chips. C(97,5) is the number of combinations of 5 non-black chips out of 97.

in total you are selecting 6 chips out of 100. C(100,6) is the number of combinations of 6 chips out of 100.

probability = C(3,1) * C(97,5)

----------------------

C(100,6)

= 3 *64446024

----------- = 0.162

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