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I don't believe the Monty Hall problem is applicable to "Who Wants to Be a Millionaire?". In "Let's Make a Deal" (the Monty Hall game show), the door you choose initially cannot be eliminated, regardless of whether it is the "right" door (with the car behind it) or a "wrong" door (with a goat behind it). By contrast, in Millionaire, randomly "choosing" one of the four answers does not prevent the 50-50 computer from eliminating it.

The probability of "choosing" the right answer is 1/4, so switching will be a losing strategy 1/4 of the time.

The probability of "choosing" a wrong answer is 3/4. The computer will then eliminate two of the three wrong answers, leaving you with the right answer and a randomly-selected wrong answer. If you initially "chose" a wrong answer, then 2/3 of the time that answer will be eliminated and the question of switching becomes irrelevant; the remaining 1/3 of the time, switching will be a winning strategy.

Switching loses: 1/4
Switching is N/A: 3/4 * 2/3 = 1/2
Switching wins: 3/4 * 1/3 = 1/4

Your student's suggestion would indeed be fantastic if and only if the 50-50 computer was not allowed to eliminate your "chosen" answer. If that were the case, switching would be even better in Millionaire than in "Let's Make a Deal" (it would be a winning strategy 3/4 of the time as opposed to 2/3 of the time).

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