Deal or No Deal?

Question

You are playing a game called “Deal or No Deal”, twenty-six cases are displayed at random with various amounts in each (from one cent to one million dollars). You select case number one and manage to eliminate all but two cases (your case and case two). One case contains one million dollars and one contains one cent. Howie Mandel ask: “Would you like to switch to case number two?” Is it to your advantage to switch to case number two?

Explain your answer.

Answer 1

I would switch. Odds of me picking the million first thing out of the 26 cases is far less likely than me narrowing down to that one case left out of the 25.

Although, seriously, if the two cases left are one cent and one million...I'd take the banker's deal. :)

Answer 2

This depends how 24 of the cases were eliminated. The wording is vague.

The chance of picking the case with a million dollars is 1/26th. The chance of any other case (including the case you didn't eliminate) is 1/26th.

If the unchosen case was the only one left just by random chance, then you have a 50/50 chance either way.

If Howie Mandel intentionally set up a situation where one of the remaining cases (the one you picked, as well) would always have a million dollars, then there's a 25/26th chance of the unpicked case having the million dollars, which means you should switch.

If Howie Mandel intentionally set up a situation where one of the remaining cases (the one you picked, as well) would always have 1 cent, then there's a 25/26th chance of the unpicked case having the 1 cent, which means you should keep the case you picked.

With the information given, there's no way to know and a 50/50 chance of either strategy working.

Answer 3

Jeffrey, trust me on this; the chances are 50% for either case to have the million dollars. There is no Monty Hall paradox here. The host never interferes with the picking of the cases that are opened. The key behind the Monty Hall paradox is the the host knows which doors have which prizes, and therefore knows which door to pick when revealing to the contestant. This can also be shown with simulation. No change in probabilities. You are just as likely to get the million whether you switch or not.

In the message board post you sent me, the Deal or No Deal case would be similar to the second case that you discussed rather than the first. However, that case does not fall under the Monty Hall paradox.

I would refer you to the discussion of this Wikipedia page on Deal or No Deal. There is a whole section on this discussion that deals with the Monty Hall paradox.

Answer 4

The wording of the question is critical. How are 24 cases eliminated? If I select one, and Howie then picks out 24 cases knowing that none of them have the $1,000,000, what are the odds that the last one he left unopened has the $1,000,000? Very high. The odds of it NOT having that $1,000,000 is 1 in 26, because I happened to have selected the case with the $1,000,000 myself! So, I'd definitely switch, since the odds of that 25th and uneliminated case having the $1,000,000 is 25/26.

Answer 5

I don't think it will make a difference

Like the lyrics in a Rush song "If you choose not to choose you still have made a choice"

You have just reduced your odds down to a single 50/50 decision. Did you pick the right case first or not.

Answer 6

I'd keep my case. Once, this guy decided to switch cases. He ended up going home with 1 cent!

Answer 7

I think I'd go with Shauna and take the banker's offer. But strictly from an odds situation, it's 50:50. Flip your mental coin, but you still have an equal chance all the way around.