Expert: Ahmed Salami - 4/27/2009

Question
Bucket A contains three different colored balls: one red, one blue, and one white.  Bucket B contains four different colored balls:  one red, one blue, one white and one black.  A blindfolded man is told to pick out a ball from bucket A, and, if the ball selected is not red, to then proceed and select a ball from bucket B.  What is the overall probability that he will select a red ball given that if he does not select a red ball on the first try from bucket A he can then make a second try and pick from bucket B?

Answer
Hi Ken,
The probability of picking a red ball in the first bucket, assuming identical balls
P(red.first bucket) = 1/3
The probability of failing to pick a red ball in the first bucket
P(not red.first bucket) = 1 - 1/3
                       = 2/3
The probability of picking a red ball in the second bucket
P(red.second bucket) = 1/4
The man ends up picking a red ball if he either picks it from the first bucket or if failing on that picks it in the second bucket i.e he either succeeds in the first trial OR fails in first AND succeeds in second trial. The probability of that happening
P = P(red.first bucket) + P(not red.first bucket) x P(red.second bucket)
 = 1/3 + 2/3.1/4
 = 1/3 + 1/6
 = 1/6
You could also have gone about it this way. Eventually, the man either picks a red ball or not and the sum of the probabilities of the two events must be equal to 1. The man fails to pick a red ball if he fails in both attempts of the first and second bucket i.e fails in the first AND fails in the second
P(no red) = P(not red.first bucket) x P(not red.second bucket)
         = 2/3 x (1 - 1/4)
         = 2/3 x 3/4
         = 1/2
Now,
P(red) + P(no red) = 1
P(red) = 1 - P(no red)
      = 1 - 1/2
      = 1/2

Hope it helps you.

Regards