Brain Or No Brain

I’m not usually inclined to watch Deal Or No Deal. It’s not appointment television for me by any means. But it is on prior to several shows that are appointment television for me, most particularly Heroes, and so I will, on occasion, watch parts of it. If you don’t know how it’s played, here’s a nice Wiki write-up.

So anyway, last night’s contestant was a big ol Southern guy, currently unemployed, living in his sister and brother-in-law’s basement. His sister, brother-in-law, and best friend were there as his supporters. He got off to a nice start, eliminating lower-valued briefcases from the board, and at one point, getting a $218,000 offer from the banker.

Now here’s the thing. After accounting for the fact that you’d probably lose roughly 35% to taxes, with $218,000, you’d clear around $140,000. I don’t know about you, but $140,000 would pay off all my debt, buy a new car, and give me a serious down payment on a house (or, if not a house, cover rent payments for four to five years) with some left over, and I’m absolutely certain I probably have more debt than our contestant. I’d have jumped so fast at the deal it would make head spins, which probably eliminates me from the contestant pool because I’d be too willing to make a deal instead of gambling and still playing.

Not last night’s Brainiac, however. He chose to keep playing, after a couple more rounds of choosing briefcases, was down to a $57,000 or so offer from the banker. There was only one high value briefcase left on the board, the $750,000 one, and approximately five low value ones. But he kept playing, eliminating several low value briefcases, until there were only three left: the $750,000, the $100, and the $10. This prompted an offer from the banker of $150,000, which after taxes would still be roughly $98,000. I’m no longer making a down payment on a house with that, but I’m still paying off all my debt, buying a new car, paying rent on a new place for a year, and having some left over.

Encyclopedia Brown’s support group was mixed. His brother-in-law and best friend were telling him to say no deal and eliminate another briefcase. But not his sister. Oh no. You could tell she wanted nothing so much in the world as for him to take the money and move the hell out of her house. When he said no deal, she plopped down in her chair, unable to watch what happened next. So what happened next? The next briefcase he picked held the $750,000, meaning his prize was either $100 or $10. He rejected the banker’s offer of $55, a split of the difference, and then walked away with $10, roughly, oh, $217,990 less than he could have had.

So why am I writing? Is it to comment on a show so idiotic that the only skill you need is to be able to identify the numbers between 1 and 26? Is it to comment on the gambling mentality that says keep playing long after the odds have shifted against you? No. It’s because of this.

As they kept eliminating briefcases, they would update and say things like “You started with a 1 in 26 chance of having a highest remaining dollar value briefcase and you now have a 1 in 20 chance!” But I don’t think this is correct. If you turned your briefcase in each round and selected a new one, yes, when you selected your new one you would have a 1 in 20 chance, but you don’t do that. So instead, you still have just a 1 in 26 chance, the original odds when you chose your original briefcase. Because your briefcase doesn’t change during the course of the game, the original odds don’t change. At least that’s how I see it intellectually.

Yet, somehow, this doesn’t seem right intuitively. By the time you are down to three suitcases, you have a one in three chance that the briefcase you initially chose has the highest remaining dollar value. Right? Or are we talking two different things? Is it that you still have the 1 in 26 chance you chose the highest remaining dollar value briefcase initially, but the odds that you have the highest remaining dollar value briefcase in your possession changes as briefcases are eliminated? Does that make sense? Help me, mathletes!