The Definitive FAQ/HOWTO on how to win a million dollars on Deal or No Deal

Background

The standard Deal or No Deal Game (henceforth "Deal") offers contestants their choice of one of twenty-six briefcases containing a number representing the amount of money they would win.

Contestants pick a case ("Their Case") and then slowly work to eliminate other cases as the "Banker's offer" to "buy their case from them" increases or decreases with the odds of the unexposed remaining amounts.

Strategy and Luck

There are several goals, and depending on your goal, you might choose different strategies, and all depend on whatever is randomly in those cases... in other words -- luck.

Goals

  1. Winning a million dollars. This FAQ will focus on this goal.
  2. Winning more than what the odds represent an average win to be.

Odds

When there are 26 briefcases, your odds of winning one million dollars are exactly 1 in 26 (less than 4%).

If you eliminate some cases, your odds of winning one million dollars are exactly 1 in 26. That's right the same number.

FAQ

Q1: But I eliminated some low numbers!!! They say my odds are higher!!!
A1: If you were able to return your original briefcase to the pool and select a briefcase, those odds would apply to you. Unfortunately you cannot. And so the original selection odds stick with you throughout the game. This is known in a more famous version as "The Monty Hall Problem".

Q2: But the offers are going up up up!!!
A2: The offers will either go up or down but the important thing is that to win the million you must reject ALL offers and end up with your original briefcase. The amount in there will be one million dollars 1 out of 26 times.

Q3: What's the best strategy?
A3: To win a million dollars you must open all 25 other briefcases, and had selected the million dollars for your suitcase. This will happen 1 out of 26 times.

Q4: What if just before the final briefcase I'm offered the offer to switch the case I chose with the case unopened on the stage?
A4: If you switch you increase your odds to 25/26. The reasoning is simple (and again this is "The Monty Hall Problem"). There were 26 cases. The odds that yours holds the million is 1/26. The odds the remaining suitcase holds the million is 1-1/26 or 25/26. Switching is always the way to go. You'll notice that's not an option.

Q5: Ok, so let's say I'm giving up on the million, how do I maximize my win?
A5: Pick a bunch of low numbers and when the offer exceeds the average of the amounts originally on the board, take it. The Interweb says around $130K. (The exact number varies with the number of $1,000,000 prizes on the board.)

Q6: It seems the first few offers are lowballs...?
A6: Yes. They want the player to stay in, and the easiest way to do that (since they can only influence the player with LOW or HIGH offers) is to offer insultingly-low offers.

Q7: Are you saying that this game is set from the very first case selection?
A7: Yes, I am.

Q8: Well I saw this one guy and he went almost all the way!!!
A8: Yes. The odds his suitcase had $1M was 1/26. If he got more than $130K he beat the odds. He would have had to remove all the rest of the high numbers to find out if he had a million.

Q9: Well there have been more than 26 shows... why did nobody win the million?
A9: Nobody eliminated 25 cases. That would be ... beating the odds by 26:1.

Q9: Are you affiliated with the show or what?
A9: No, I'm not. I once took fourth-grade math and they taught us division. I realize that this is now a lost art and all, but that's life.

Ehud