Deal or No Deal and Monty Hall Problem?

Question

For the rules of these games, try searching them at Wikipedia.

My questions:

In the Monty Hall Problem in which the host always opens a door hiding a goat, the chances to win a car is greater if the player chooses to switch doors - that is 2/3 to 1/3. In "Deal or No Deal," supposed the contestant is left with 2 briefcases; one he hasn't opened yet and the other briefcase is the one he initially picked at the beginning of the game. And supposed that the only amounts remaining are $1 and $1M. If the contestant yells "No Deal", what are his chances of winning the million bucks? 1/2 or still 1/26?

Supposed the contestant has four briefcases remaining: three he hasn't opened yet and the other briefcase is the one he initially picked at the beginning of the game. And supposed that the amounts remaining are three small amounts and $1,000,000. If the contestant says "No Deal" and has to pick a briefcase to open, is it right to say that he has a 75% chance of NOT opening the $1M?

Answer 1

1/2 and yes. The boxes are independent so your probability of picking any cash amount is 1/number of boxes remaining.

Answer 2

Every time he opens a case it is an independent event. The chances on that case containing a particular value is 1 over the number of possible choices.

The Monty Hall problem has several factor that don't apply to Deal or No Deal. For instance Monty Hall knows where the prize is, and always opens a door that doesn't contain the prize, which makes the difference in the statistical probability. On Deal or No Deal the host does not know the location of the prize and does not remove a bad possibility from the playing field for you.

I'm not a big fan of Wikipedia references but there is a quite good explanation of the Monty Hall problem and how that particular circumstance leads to a dependent events.

Answer 3

If there are only 2 cases remaining, he has a 50% chance of selecting 1million dollars (as long as that case is left), not 1/26 anymore. This is because he has knowledge of the new sample size and what is remaining.

If there are 4 cases remaining and he can only choose one case, there is a 75% chance he selected the wrong case.

Watch how the "banks offer" varies. It is based on how many cases are remaining, the total value of all the cases, and the average value of each case. *The numbers work out in favor of the house... just like a casino.