WHO WANTS TO BE A MILLIONAIRE?

MAY 4, 2000

 

The Seventh Annual Hudson River Undergraduate Mathematics Conference on April 8, 2000 at Vassar College included 145 talks by some 450 students and faculty from 87 institutions. Peter Hilton, in the invited address, “opened many students’ eyes to the fact that math’s beauty [lies] in the combination of intrigue and importance,” according to participant Eric Katerman, Williams ’02.

LARGE PRIMES near powers of ten are reported by Marvin Ray Burns. For example:

10000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000000000000000000000000000000000000000
00000000000000000000000000000000000000000002569

(one hundred quinsexagintillion, two thousand, five hundred, sixty nine).

QUESTIONABLE MATHEMATICS. On April 21, The Christian Science Monitor reported that “In 1952, Democrats had a 58-point advantage among Southern white men. . . . But by 1992, Republican had a one-point advantage. . . That means there was a 59-point shift of men away from the Democrats.”

Of course, it actually means just a 29.5-point shift, e.g, from 79% (58 points above 21%) to 49.5% (1 point below 50.5%).

Readers are invited to send in more examples of questionable mathematics.

OLD CHALLENGE (Joe Shipman). On ABC TV’s “Who Wants to be a Millionaire,” after winning $250,000, you are guaranteed to keep $32,000, and you go on to $500,000 and $1,000,000 questions. After hearing a question, you can answer correctly and win the new amount, walk away with your previous winnings, or answer incorrectly and leave with $32,000. How sure should you be of your answer to the $500,000 question to answer (to maximize your expected winnings). Assume no “lifelines” (opportunities for special outside help) remain.

ANSWER (Al Zimmermann). Somewhere from about 22.5% to about 46.5%, probably closer to the latter. The answer depends on the value of the opportunity to go on to the $1,000,000 question if you get the $500,000 question.

Suppose you’ve made it to the $1,000,000 question. If you think your chance p of getting it right is less than 117/242 or about 48%, you should walk away with your $500,000; if p > 117/242, then you should answer and your expected winnings are $32,000 + $968,000p.

Now it turns out in the first case that you need to be at least 109/234 or about 46.5% sure of your answer to the $500,000 question in order to answer it. In the second case, you need only be 109/484p sure of the $500,000 question in order to answer. For example, if you felt sure (p=1) of getting the $1,000,000 question you’d need only be 109/484 or about 22.5% sure of your answer to the $500,000 question. If you felt 50% sure (p=1/2) of getting the $1,000,000 question, you’d need to be 109/242 or about 45% sure of the $500,000 question to make it worth answering. This or worse seems the more likely scenario, so you should probably be at least 45% or 46% sure to answer the $500,000 question.

FOLLOW-UP NEW CHALLENGE (Al Zimmermann). What should your strategy be if the $500,000 question is: “With how much money will you leave here tonight:

(A) $32,000 (B) $250,000 (C) $500,000 (D) $1,000,000”?

 

Send answers, comments, and new questions by email to [email protected], to be eligible for Flatland and other book awards. Winning answers will appear in the next Math Chat. Math Chat appears on the first and third Thursdays of each month. Prof. Morgan’s homepage is at www.williams.edu/Mathematics/fmorgan.

THE MATH CHAT BOOK, including a $1000 Math Chat Book QUEST, questions and answers, and a list of past challenge winners, is now available from the MAA (800-331-1622).

Copyright 2000, Frank Morgan.