Six balls are drawn without replacement from a box with 5G 7

Six balls are drawn without replacement from a box with (5G, 7R, 8B); G=green ball etc. Find P(2 of each color) P(at least one green) P(at least one of each color)

Solution

a)

There are 20C6 ways to choose any 6.

There are (5C2)(7C2)(8C2) ways to choose 2 of each color.

Thus,

P(2 of each color) = (5C2)(7C2)(8C2) / 20C6 = 0.151702786 [answer]

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b)

P(at least one green) = 1 - P(no green)

There are 15C6 ways to choose 6 non-greens.

Thus,

P(at least one green) = 1 - P(no green)

= 1 - 15C6 / 20C6

= 0.870872033 [answer]

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c)

There are 13C6 ways to choose 6 non reds.

There are 12C6 ways to choose 6 non blues.

There are 15C6 ways to choose 6 non greens.

Thus,

P(at least one of each color) = 1 - (13C6 + 12C6 + 15C6)/20C6

= 0.802760578 [answer]