Three balls of the same color?

Probability Level 4

A bag contains 100 red balls, 10 green balls, and some blue balls.

How many blue balls would minimize the probability that you could pick 3 balls without replacement and they would all end up being the same color?

Assume that there are no other colors in the bag.


The answer is 95.

Geoff Pilling by Geoff Pilling , 53, USA

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1 solution

Geoff Pilling
Sep 2, 2016

The probability they will be the same color is given by:

P ( n ) = ( 100 3 ) + ( 10 3 ) + ( n 3 ) ( 110 + n 3 ) \large P(n) = \frac{\binom{100}{3} + \binom{10}{3} + \binom{n}{3}}{\binom{110 + n}{3}}

This is minimized for n = 95 n = \boxed{95}

1 Helpful 0 Interesting 0 Brilliant 0 Confused

A curious result. Before making the calculation I guessed at something in the low 30's, i.e., close to the geometric mean of 100 and 10. So much for guessing...

Brian Charlesworth - 4 years, 9 months ago

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Haha... Yeah, I love answers that aren't intuitive... :0)

Geoff Pilling - 4 years, 9 months ago

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How did you solve for n n ?

Pi Han Goh - 4 years, 9 months ago

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@Pi Han Goh – I suppose you could do P ′ ( n ) = 0 P'(n) = 0 , but admittedly I didn't go that route, I just kept plugging in different values for n n until I found a local minumum... :-/ And now I know you, Pi, your going to ask, "How do you know there isn't another local minimum that's smaller?" ;^)

Geoff Pilling - 4 years, 9 months ago

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