Problem 23 · 2014 Math Kangaroo
Hard
Counting & Probability
careful-counting
There are 9 kangaroos called the Greatkangs. Each is coloured either white or black. If three Greatkangs meet by chance, the probability that none of them is white is exactly two thirds. How many Greatkangs are black?
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Answer: E — 8
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Hint 1 of 2
'None of three is white' means all three are chosen from the black ones.
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Hint 2 of 2
Set the probability of an all-black trio equal to 2/3 and solve for how many are black.
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Approach: ratio of all-black trios to all trios
- With b black out of 9, P(all three black) = C(b,3)/C(9,3) = C(b,3)/84.
- Set C(b,3)/84 = 2/3, so C(b,3) = 56.
- C(8,3) = 56, so b = 8 Greatkangs are black.
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