help on probability

Quote:

Originally Posted by math8553

In a tennis game once the score has reached "deuce"play continues until one player has a lead of 2 points

Mike and Bob are playng tennis and the score reached deuce.Suppose each point is won by Mike with probability 1/4(and otherwise by Bob)and the outcome of each point is independent of all other points.

If "x" is the probability that Mike wins,"u" is the conditionl probability that Mike wins given that he wins the first point and "v" the conditional probability Mike wins given that he loses the first point
How do i use the theorem of total probability to show that

4x=u+3v
4u=3x+1
4v=x

how can i find the probability that Mike wins the match?

If you draw a tree diagram it's easy to see where the first equation has come from. Let the first two branches be "Wins first point" and "Doesn't win first point". Let the next two branches (from each of the first branches) be "Wins game" and "Doesn't win game".

Then you should see that $\frac{u}{4} + \frac{3v}{4} = x \, ....$

I can't readily see where the other two equations come from ..... But if you solve them simultaneously you get x = 1/10 (which is the correct answer).