As promised, I chose to write about the Monty Hall problem this week because it is one of the most confusing probability problems to the average person. The example I’ll be using today is different from the original problem because I think that it is a clearer example and is also far more modern.
Consider the game show “Deal or No Deal,” in which the contestant picks one of twenty-five briefcases (all with different values ranging from $0.01 to the $1,000,000 grand prize) to keep with him/her and remains unopened until the end of the game. The contestant then opens one of the cases he/she did not choose and this continues until the final two cases—his/hers and one other. During all this, the contestant receives bids from the host to try to sell the case without knowing what is in it (the host acts as if the bids use a complex formula but in reality it is just the average of the remaining cases because each remaining cash value is equally likely to be in the contestant’s case). This bidding process builds a strong relationship between the contestant and the case, and therefore bias which will be very important for later in the example.
Now, let us assume that the contestant has rejected every bid and it is now down until there are two cases remaining—the contestant’s and one other. Let us also assume that one of the cases has $0.01 and the other has $1,000,000. At this point, Howie Mandel offers one last potential move before the big reveal. He always offers the contestant the opportunity to swap cases before finding out what amount is in each case. What should the contestant do?
Of course, he/she should switch! There is a 50% chance of there being $1,000,000 in each of the two cases (let’s call them case 1 and case 2). That is fairly obvious, but what many people unfortunately fail to consider is the fact that case 1 (the contestant’s case) has only a 4% chance of having the $1,000,000, the same as any other one of the 25 cases. They chose the case from the original pool of 25 whereas they can now choose the case from the pool of 2 and have a 46% better chance of winning. (I digress that the safest move would be to sell the case and walk away with $500,000 but let’s assume they like to live dangerously.)
Honestly, the entire layout of the game is brilliant. The contestant becomes so incredibly biased toward the case that he/she is extremely unlikely to sell and is forced into terrible odds. I also love the way that the host almost taunts the contestant by using the average to calculate the bid (which obviously implies that every remaining value is equally likely to be in the contestant’s case) but a lot of viewers don’t realize that it is just the average. But now you won’t make the same mistake that the contestant made and you’ll switch cases. Thanks for reading!
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