# The Deal with ‘Deal or No Deal’

I just saw the hit game show ‘Deal or No Deal‘.  It wasn’t the first time, but this episode had a contestant with folksiness to rival Palin, so I was entertained and kept watching.

But is there any gamesmanship to the ‘Deal or No Deal’ gameshow?  The short answer is: No.

The show begins with the contestant choosing a briefcase that contains a number that represents a real monetary amount.  The case is chosen from a group of 26 cases, with the monetary amounts ranging from a penny to a million dollars.  Recently, to up the suspense, the show has removed some of the lower amounts of money and replaced them with more million dollar cases.

The show I saw had 8 of the 26 cases carrying the million dollar value.  So when the contestant makes the initial selection, there is a slightly less than 1/3 chance of picking a million dollar case.  This case is then set aside.

The contestant then proceeds to pick other cases which are immediately opened, revealing the monetary amount they represent.  These cases are removed from the pool of cases.  After a few cases have been removed, the contestant is offered a sum of money to stop playing.  If many of the cases that have been removed were low in value, i.e. most of the million (and other high value) cases remain, then the offer will be closer to the high value cases.  If many of the high value cases have been removed, then the offer will be closer to the lower values.  Usually the value is somewhere in the middle.

These offers are made periodically when there are many cases remaining and are made after every case for the last few.  If you go all the way to the end, then you receive whatever value is in the case you initially selected.

If winning the big prize is the goal, however, all the offers are completely irrelevant.  At the outset the case the contestant chooses has a 1/3 chance of containing the big prize.  This doesn’t change throughout the game.  Let me explain why:

The rest of the cases have the same approximate ratio of million dollar values to non-million dollar values, which the contestant chooses to open randomly.  Therefore most of the time (logically speaking and whenever I watched) this ratio stays constant all the way to the end of the game.  2 cases out of the last 6 were million dollar cases in the episode I just saw.

Of course the possibility exists that the contestant will choose all of the lower value cases such that only million dollar cases remain and hence the case he or she initially chose will necessarily be a million dollar case.

However, imagine this analogous situation.  Try to pick all the cards other than Jack, Queen, King and Ace out of a shuffled deck without looking.  What will happen is that a selection of cards will be chosen irrespective of value, randomly, leaving approximately the same ratio of face cards to non-face cards remaining  (Go try it if you don’t believe me).  The chances of picking only the low values are very small.  Deal of No Deal has been on for years here in the USA and this has never happened.  The recent, and only, million dollar winner still had to choose on the last remaining case. So this part of the game has little ultimate impact upon knowing whether or not you have selected a million dollar case.

Secondly, since the cases are opened randomly during the show, no Monty Hall-like insight can be gained as to whether or not a winning case was initially selected.  Therefore the initial probability of 1/3 remains unchanged throughout the show and all the song and dance of selecting and opening the cases is a red herring (though it is top notch song and dance provided by Mr. H. Mandel and models).

This leaves the contestant in the position of deciding whether or not to accept the offer made to stop playing part way through the game without any new information.  Since the ratio of remaining monetary values remains somewhat constant, the offer made to buy the contestant out of playing should remain somewhat stable for most of the game.  It appears however, according to Wikipedia, that the initial offers are kept artificially low to build suspense, but at the end the offers are where the mathematicians say they should be.

The decision then comes down to how badly the contestant wants/ needs the money.  If the money offered to stop playing becomes large enough to significantly, to the contestant’s mind, make a big difference, he or she will likely take the money rather than take the 2/3 chance of winning significantly less.  This is what happened during the episode today: after it was made known late in the game that a sponsor was going to make a matching donation to a national charity the lady supported, she became too afraid of losing the large amount of money that was already offered, even though she said she wanted to go till the end.

In the end, the deal with ‘Deal or No Deal’ is that it is a great deal for those who get to play.  However, it is not much of a game.  The only trick is to get yourself on the show and after that how much you take home is up to luck.