Marilyn answers a lottery question

Ask Marilyn
Parade, April 10, 2005
Marilyn vos Savant.

A reader poses the following question:

My wife and I attended a "reverse raffle," in which everyone bought a number. Numbered balls were then drawn out of a bin one at a time. The last number would be the winner. But when the organizers got down to the last couple of dozen balls, they discovered that some numbered balls had been overlooked. So they added those balls to the bin and continued the drawing. Didn’t the added balls have a much better chance of winning?

Marilyn responds, "Yes, they did. But because everyone had an equal chance of his or her numbered ball being one of those overlooked, the last-minute addition made no difference to anyone’s chance of winning. The raffle was still fair."

Marilyn's answers to probability problems often stir up controversy, and this was no exception. The discussion continues in the column below.

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Ask Marilyn.
Parade, June 5, 2005
Marilyn vos Savant.

A reader had this objection to Marilyn's answer:

Marilyn: I disagree with your answer. Participants whose balls were left out had a higher likelihood of winning. Regardless of whether they had a fair chance of being overlooked, the raffle was not mathematically fair. Assume there were 20 participants. The odds of winning should be one in 20 throughout the game for each contestant. Put 15 balls in a jar and withdraw 10. Then add the missing five.

The first 15 balls had a two-in-three chance of “not winning” until the five balls were added. The missing five balls had a zero chance of “not winning” during that time, then had a one-in-10 chance of winning after they were added. Only the five balls that were in the bowl the entire time had a one-in-20 chance of winning.

Marilyn says her original answer was correct, and asks the reader to "consider a scenario in which the added balls were withheld (on purpose) instead of overlooked." She says. "Your explanation works in that case. So it cannot work in the case when the added balls were merely overlooked."

DISCUSSION QUESTIONS:

(1) Do you understand Marilyn's last response?

(2) The reader is actually giving conditional probabilities. Do you agree with their values?

(3) Now consider the reader's set-up, under Marilyn's original assumption that each ball had an equal chance of being overlooked in the first stage. Thus, there are 20 balls, and 15 are initially selected at random and placed in the bin. Now 10 are drawn one at a time at random from the bin. At this point, the 5 balls originally omitted are added to the bin. Then balls are drawn one at a time at random from the bin. Looking at the whole process, does each ball have a one-in-20 chance of being the last ball in the bin?

(4) Upon realizing the correct answer to Marilyn's problem a chance news reader commented that this suggested that the famous 1970 Vietnam lottery was fair also. Recall that in this lottery 366 balls with the possible dates in a year were put in a bowl and mixed up. Then the balls were drawn out one at a time and the dates on the balls determined the order in which drafties would be called up. It was estimated that those with birthdays on the last third of the balls drawn would not be called up at all. Å statistical analysis suggested that the balls were not well mixed and as a result those born in the early months were sigfinicantly more likely to be called up than those born in the later months. But our reader said that since are birthdays are random the lottery is still fair. Do you agree?

P.S. You can read the New York Times account of the statistical challange of this lottery here and a nice article by Norton Starr on how to use this lottery in a statistical class.