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# Deal or No Deal: Gambling With Math!

I’ve caught the game show Deal or No Deal twice over the last couple weeks and it’s driving me crazy! It’s a mathematical game but the contestants don’t play it mathematically. It’s getting to the point where I can’t watch it anymore!

How the Game Works

If you haven’t seen the show, it’s a pretty simple concept. At the beginning of the game the contestant selects a briefcase containing an unknown sum of money. Throughout the game a “banker” offers the contestant a flat sum to walk away from the game (Deal) or give up the money for a chance to open their own case (No Deal).

During each round the number of cases to open changes as does the offer from the bank. The smallest amount is \$0.01 and the largest amount is \$1 million.

Decision Making

First I watched an episode where the player had 2 cases remaining with the following dollar amounts:

• \$0.01
• \$10,000

The offer from the bank was \$5,500. What would you do? Unfortunately she didn’t take the offer, opened her case and was the first player to leave the show with one penny. Would it change your opinion if earlier she had already given up many higher offers, the highest at \$207,000?

I also caught a variation of the game show on Oprah. The two final cases contained these amounts:

• \$5,000
• \$75,000

The offer from the bank was \$50,000. Oprah chose to open the case…. and guess what it contained? Of course it only had the lower amount in it.

Expected Outcome

This expected outcome can easily be calculated in this game. The formula for expected outcome is:

A simple example from wikipedia [2] is the expected value when you roll a die:

In the examples above, the expected outcome for \$0.01 and \$10,000 is \$5,000.005. Of course there isn’t a half penny, but the offer of \$5,500 is almost \$500 above the expected outcome.

In the second example, the expected outcome for \$5,000 and \$75,000 is \$40,000. The offer of \$50,000 was \$10,000 above the expected outcome!

What I Would Do

It pains me to watch this show. In both cases I would take the deal. In general I don’t have much risk aversion, but this type of game is all math to me. If the offer from the bank is more than the expected outcome, I will walk. I have discussed it with various family members and found some opinions are very different from mine.

I consider the money in each round already mine. In the first example, if you came up to me on the street and asked me to bet \$5,500 to win either a penny or \$10,000, I would think you are nuts! My money has no business participating in that kind of expected loss.

Other Considerations

Of course, it’s really easy for me to sit at home yelling at the t.v. Under the lights, with adrenaline pumping, and playing for real, I might make a different decision. I really can’t say for sure. Not to mention the family members you bring along that try to convince you which way to go.

The amount of money also has some bearing on whether following a mathematical equation makes sense. In the first example people might be more willing to risk \$5,500 than they would \$40,000. And Oprah has a lot of money, so her threshold is probably much higher than other people.

Action Plan

I’d like to think that I know I would adhere to mathematical principles in a mathematical game. It’s probably also the same reason I wouldn’t ever be invited to be on the show, because they would know exactly what I would do in any given situation.

This isn’t to say that I don’t gamble, I do… more about that some other day!

Deal or No Deal?