What is the Probability of Drawing a White Ball from the Second Urn?

[Debdut]

An urn contains 10 white and 3 black balls. Another urn contains 3 white and 5 black balls. 2 balls are drawn at random from the first urn and placed in the second urn and 1 ball is drawn at random from the second urn. What is the probability that it is white?

Is the answer 59/130

 

[micromass]

Please tell us why you think that.

 

[xiavatar]

Check your arithmetic debdut.

 

[Ray Vickson]

Check your arithmetic debdut.

I also get 59/130. Nevertheless, the OP should show his/her work.

 

[Debdut]

I don't know whether I have done it correctly or not...

There are 4 combinations of picking 2 balls from 1st urn (White → W, Black → B)

WW → P = 10/13 x 9/12 = 15/26
BB → P = 3/13 x 2/12 = 1/26
WB → P = 10/13 x 3/12 = 5/26
BW → P = 3/13 x 10/12 = 5/26

If WW was picked, then balls in 2nd urn = 5W 5B, Then probability of picking white ball = 5/10
If BB was picked, then balls in 2nd urn = 3W 7B, Then probability of picking white ball = 3/10
If WB was picked, then balls in 2nd urn = 4W 6B, Then probability of picking white ball = 4/10
If BW was picked, then balls in 2nd urn = 4W 6B, Then probability of picking white ball = 4/10

Thus total probability of picking white ball from 2nd urn = (5/10 x 15/26)+(3/10 x 1/26)+(4/10 x 5/26)+(4/10 x 5/26) = 59/130

 

[HallsofIvy]

Yes, that is correct.