Three boys play a game as follows. They put three white balls and a red ball in a box. Andy, Bruce, and Charles, in this order, each choose a ball at random from the box, without replacement. Whoever gets the red ball wins. If none of the three draws the red ball, nobody wins. Which one of the three boys has the largest probability of winning?
Since the balls aren't being replaced, I thought Charles should have the highest probability, but it seems that they all have equal chances of winning. How is this possible?
Best Answer
A winning: $$P(R_A)=\frac{1}{4}.$$ B winning: $$P(W_A)P(R_B|W_A)=\frac34 \cdot \frac13 = \frac14.$$ C winning: $$P(W_A)P(W_B|W_A)P(R_C|W_A\cap W_B)=\frac34 \cdot \frac23 \cdot \frac12=\frac14.$$