Two Identical Urns Contain Balls

Question

Examlex · Accepted Answer

The Answer is (a) Half the time we select the first urn, in which case the probability that the two balls are both red is

( 6 / 9 ) ( 5 / 8 ) = 5 / 12

. Half the time we select the second urn, in which case the probability that the two balls are both red is

( 5 / 13 ) ( 4 / 12 ) = 5 / 39

. Therefore the answer is

\frac { 1 } { 2 } \cdot \frac { 5 } { 12 } + \frac { 1 } { 2 } \cdot \frac { 5 } { 39 } = \frac { 85 } { 312 }

(b) Let

F

be the event that the first ball is red, and let

S

be the event that the second ball is red. We are asked for

p ( S \mid F )

. By definition, this is

p ( S \cap F ) / p ( F )

. In part (a) we found that

p ( S \cap F ) = 85 / 312

. By a simpler calculation, we see that

p ( F ) = \frac { 1 } { 2 } \cdot \frac { 6 } { 9 } + \frac { 1 } { 2 } \cdot \frac { 5 } { 8 } = \frac { 31 } { 48 }

Thus the answer is

\frac { 85 / 312 } { 31 / 48 } = \frac { 170 } { 403 }

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