PROBLEM 8E

An urn contains 17 balls marked LOSE and 3 balls marked WIN. You and an opponent take turns selecting a single ball at random 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 probability that your opponent wins the game on his second draw.

(c) If you draw first, what is the probability that you win? Hint: You could win on your second, third, fourth, . . . , or tenth draw, but not on your first.

(d) Would you prefer to draw first or second?Why?

Answer :

Step 1 of 4 :

We have an urn contains 17 balls marked LOSE and 3 balls marked WIN. You and an opponent take turns selecting a single ball at random 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 first, we need to find the probability that you win the game on your second draw.

Here we consider Y : you draw W first, O : opponent draws W first and

Y : you draw W second.

Then,

P(YOY) = P(W W W )

Where W is win.

P(YOY) =

P(YOY) =

Therefore the probability that you win the game on your second draw is.

Step 2 of 4 :

b).

If you first, we need to find the probability that your opponent wins the game on his second draw.

P(YOYO) = P(2 Win,1 Lose , 1 Win)

P(YOYO) =

P(YOYO) =

P(YOYO) =

Therefore the probability that your opponent wins the game on his second draw is .