an urn contains 20 balls of which 8 are yellow and 12 are re

an urn contains 20 balls of which 8 are yellow and 12 are red. if six balls are drawn without replacement, what is the probability that

a) All six are the same color?

b) at least four are of the same color?

Solution

There are 20C6 = 38760 ways to choose 6 balls.

a)

If all are yellow, there are 8C6 = 28 ways.
If all are red, there are 12C6 = 924 ways

Hence, there are 924 + 28 = 952 ways.

Thus,
P(all same) = 952/38760 = 0.024561404 [answer]

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b)

There are 6 cases here:

4Y, 2R has (8C4)(12C2) = 4620 ways
5Y, 1R has (8C5)(12C1) = 672 ways
6Y, 0R has (8C6)(12C0) = 28 ways
4R, 2Y has (8C2)(12C4) = 13860 ways
5R, 1Y has (8C1)(12C5) = 6336 ways
6R, 0Y has (8C0)(12C6) = 924 ways

A total of 26440 ways.

Hence,

probability = 26440/38760 = 0.682146543 [answer]