# Math Help - Probability question

1. ## Probability question

An urn contains 17 balls marked LOSE and 3 balls marked WIN. You and an opponent take turns selecting at random a single ball from the urn without replacement. The person who selects the third WIN ball wins the game. It does not matter who selected the first two WIN balls.

a) If you draw first, find the probability that you win the game on your second draw.
b) If you draw first, find the probablility that your opponent wins the game on his second draw.
c) If you draw first, what is the probability that you you win? Hint: You could win on your second, third, fourth, ..., or tenth draw, not on your first.
d) Would you prefer to draw first or second? Why?
Thanks

2. Originally Posted by Amy
An urn contains 17 balls marked LOSE and 3 balls marked WIN. You and an opponent take turns selecting at random a single ball from the urn without replacement. The person who selects the third WIN ball wins the game. It does not matter who selected the first two WIN balls.
a) If you draw first, find the probability that you win the game on your second draw.
b) If you draw first, find the probablility that your opponent wins the game on his second draw.
c) If you draw first, what is the probability that you you win? Hint: You could win on your second, third, fourth, ..., or tenth draw, not on your first.
d) Would you prefer to draw first or second? Why?
Thanks
There are ${{20} \choose 3}=1140$ ways to draw these balls from the urn.
a) There is only one way for you to win on the second draw:WWWLLL… .
b) How many ways can we arrange WWL? Now add a W.
c) The number of ways that you win on your second, third, fourth, ..., or tenth draw
is : $\sum\limits_{k = 0}^8 {\frac{{\left( {2 + 2k} \right)!}}{{2\left[ {\left( {2k} \right)!} \right]}}}$.
Can you explain part c?