You play a game where there are three white marbles and one

You play a game where there are three white marbles and one black marble in a bag. You get to draw one marble at a time. If you draw the black one, you lose. If you don’t, you keep the white marble, and draw again. If you get all three white marbles, you win. (show work)

A. What is the probability of winning?

B. What are the odds of winning? Write as a reduced ratio of whole numbers.

C. Is this game fair? i.e. is the probability of winning and losing the same?

D. If you bet $1 to play, what should the “house” bet to make it fair? i.e. how much money should you receive if you win the game?

Solution

A.

P(win) = P(white first) P(white second) P(white third)

= 3/4 * 2/3 * 1/2

= 1/4

= 0.25 [ANSWER]

*********************************

B.

As

odds = [1 - P(win)]/[P win]

= [1-0.75]/0.25

= 3 [ANSWER]

******************************

No. P(losing) = 0.75, P(win) = 0.25.

******************************

D.

Let x = the winning prize.

If you lose, you lose $1.

If you win, you win x - 1.
Then, to make it fair,

(-1)P(losing) + (x - 1)P(win) = 0

-0.75 + (x - 1)(0.25) = 0

x = 4 [ANSWER]