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## Tuesday, April 10, 2007

### More Deal or No Deal

At the request of Dan, I have done a statistical breakdown of a full episode of Deal or No Deal. This will be an uncharacteristic post for me, because although my focus is mathematics education, I rarely if ever include a lot of numbers in my posts. Some of my readers, if I have any readers that is, may want to skip this one.

I chose to analyze an episode of the UK version of the show, primarily because I found a fansite that had done most of the work for me, but also because there's only so much Howie Mandel I can stand. The British version is identical in theme, but differs in a few details. There are 22 boxes, instead of the 26 in the US. The currency values in pounds are as follows:

1p 1000
10p 3000
50p 5000
1 10,000
5 15,000
10 20,000
50 35,000
100 50,000
250 75,000
500 100,000
750 250,000

I have decided that it makes the most sense to look at the game from the banker's perspective. The money is really his to lose, since the contestant is guaranteed to win something. The total money on the board at the start is 565,666.61, therefore the expected average value of the game is just the arithmetic mean of 25,712.12. Following the minimax theory of game theory, it should be the bankers goal to keep the contestant below that average value.

In this respect, the banker is really more of a psychologist than mathematician. Like a poker player upping the ante, his goal is to keep his opponent in the game for as long as possible. In the UK version, there are sometimes multiple contestants per show, ala Millionaire. Since the banker pays out the 25,712.12 average per contestant, not per episode, it is in his interest to limit the number of contestants, thereby increasing the length of each game. In the US, there is only one contestant per show, so the goal is simply to fit the time alloted.

I first began by tallying up the winnings from the first 20 games of the inaugural season to see how the banker did. His actual average payout was only 18,720 , which is roughly 75% of the expected average payout. So he's doing pretty good. I then chose a game in which the contestant was taken the distance and ended up passing on a deal that exceeded his/her ultimate winnings. Madie from November 15, 2005 satisfied these criteria. I'm not going to list each turn here, but you can find the breakdown in the link above. Instead I will verbally summarize.

Round 1: After eliminating 5 boxes, the banker offers 6,900.
Odds of beating the deal: 6/17 or 35%
Odds of beating expected value: 3/17 or 18%
No Deal

Round 2: After eliminating 3 more boxes, the banker offers 1,600.
Odds of beating the deal: 6/14 or 43%
Odds of beating expected value: 3/14 or 21%
No Deal

Round 3: After eliminating 3 more boxes, the banker offers 4,800.
Odds of beating the deal: 4/11 or 36%
Odds of beating expected value: 1/11 or 9%
No Deal

Round 4: After eliminating 3 more boxes, the banker offers 14,800.
Odds of beating the deal: 3/8 or 38%
Odds of beating expected value: 1/8 or 13%
No Deal

Round 5: After eliminating 3 more boxes, the banker offers 28,000.
Odds of beating the deal: 1/5 or 20%
Odds of beating expected value: 1/5 or 20%
No Deal

Round 6: After eliminating 3 more boxes, the banker offers 4,800.
Odds of beating the deal: 0
Odds of beating expected value: 0
Deal

In this game, the contestants box contained 5 and the final open box had 15,000.

Glancing at the other games, it appears as though the banker chooses his "deals" so that the contestant has a 30% chance or better of beating it, thereby continuing the round and building excitement. Once it is apparent that the contestant has "lost," the deals do indeed begin to drop off. There is clearly no hard and fast rule being used, and some huge surprises occur. I suspect an analysis of a poker shark would be more appropriate here. I'm afraid I don't play well enough for that.