Conditional Dependent Probability

[lilcoley23@ho]

My questions I am looking at is:

We have twelve balls, four of which are white and eight are black. Three blindfolded players, A, B, and C draw a ball in turn, first A, then B, then C. The winner is the one who first draws a white ball. Assuming that each black ball is replaced after being drawn, find the ratio of the chances of each player.

I do not have any background on dependent proability with replacement, only without.

When making a probability tree with replacement would it look like this:

...PlayerA....Player B...Player C
...- ......- ......-
...- - - ...... - - -......- - -
...- - - -......- - - - ....- - - -
...b...w......b...w...b...w
...8/12...4/12....8/13...4/13...8/14...4/14

Please ignore the dots, it's the only way I could get my probability tree to look right.

My logic is that player A has an advantage because he's going first. So to reduce player 2's chances I added one to the sample set to symbolize that a turn had already been taken. Am I right in my thinking?

 

[mathman]

For the first round, A has a P of 1/3, B has a P of (2/3)1/3, while C has a P of (2/3)²1/3. For each subsequent round, the ratio of their chances are the same. Thus their probabilities remain in the ratio 9:6:4.
So P(A)=9/19, P(B)=6/19, P(C)=4/19.

 

[lilcoley23@ho]

Thank you so much! That makes so much more sense then what I was trying to do! Is there a way for me to rate your response as AWESOME!