There are 25 balls in a box, each of which is red or blue. If two balls are chosen at random, the chance that the two balls are the same color is the same as the chance that the two balls are different colors. What is the product of the number of red balls and blue balls?
Note: If you think there are 5 red balls and 20 blue balls, input your answer as 100.
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If the probability of getting two balls of the same colour is the same as having two balls of different colours, then both probabilities are 0.5 (complementary events).
If the number of red balls is n, then the number of blue balls is (25 - n).
The probability of having two balls of the same colour:
p ( R , R ) + p ( B , B ) = 2 5 n × 2 4 n − 1 + 2 5 2 5 − n × 2 4 2 4 − n = 0 . 5
2 n 2 − 5 0 n + 6 0 0 = 3 0 0
n 2 − 2 5 n + 1 5 0 = 0
( n − 1 5 ) ( n − 1 0 ) = 0
This means, that we have either 15 red balls and 10 blue balls or vice versa.
In either case, our answer should be:
1 5 × 1 0 = 1 0 × 1 5 = 1 5 0