Problem:
An urn contains 4 green balls and 6 blue balls. A second urn contains 16 green balls and N blue balls. A single ball is drawn at random from each urn. The probability that both balls are of the same color is 0.58. Find N.
Solution:
The event "both balls have the same color" is the union of two disjoint events, "ball 1 and ball 2 are both green" and "ball 1 and ball 2 are both blue." Because the selections from the two urns are independent, the probability that the two balls are the same color is
P( ball 1 green )⋅P( ball 2 green )+P( ball 1 blue )⋅P( ball 2 blue )=104⋅16+N16+106⋅16+NN=.58
Multiplying by 100(16+N) yields 40⋅16+60⋅N=58(16+N) which reduces to N=144.
The problems on this page are the property of the MAA's American Mathematics Competitions