Question

An urn contains 5 blue and 7 gray balls. If two (2) balls are chosen at random, one after the other, without replacement

If this experiment of choosing two (2) balls from the urn were repeated many times over, what would be the expected value of the number of blue balls?

Answer 1

Number of blue balls = 5

Number of grey balls = 7

Total number of balls = 5 + 7 = 12

P(0 blue balls) = P(first is gray and second is gray)

= 7/12 x 6/11

= 7/22

P(1 blue ball) = P(first is blue and second is grey) + P(first is grey and second is blue)

= 5/12 x 7/11 + 7/12 x 5/11

= 35/66

P(2 blue balls) = P(first is blue and second is blue) = 5/12 x 4/11

= 5/33

Expected number of blue balls = Sum of (Number of blue balls and corresponding probability)

= 0x7/22 + 1x35/66 + 2x5/33

= 5/6

= 0.8333