Scenario:

You are a game contestant in Deal or No Deal. You pick a prize, and eliminate all prizes except one more (so two prizes remaining), and you know that the top prize (the million dollar prize) is still there.

Is it better to switch your prize, or keep the one you've got?

The answer is, switch! This is a similar scenario to the Monty Hall problem that has intrigued mathematicians for years. The solution is actually pretty simple when you think about it.

When you pick one prize out of the 26 available, chances are you have not picked the top prize. The chances of picking such a prize is 25/26, where picking the top prize is 1/26.

Then, if all other prizes are eliminated, and you know the top prize is still there, then given the fact that its very unlikely you picked the top prize, you should obviously switch. This will almost always result in more money (except in the odd chance you have the top prize).

You can even make the problem more intuitive by imagining a game with a million prizes. You initially have a one-in-a-million chance of picking the top prize, which is very very unlikely. If someone then removes all the other prizes except one, and you know the top one is still out there, then obviously you'd want to switch. Because the chances you are already holding that million is very very slim.

However, you have to make sure you don't constrict the scenario to two prizes. For example, if you only compare scenarios where you are left with the million dollar prize and one cent prize, it doesn't matter whether you switch or not. The chances of getting either is equal. This is because you are constricting the probability path to just two exact prizes (which only has two possible paths). However, if you only have one locked prize (such as the top prize), then you have multiple "cases" that result in a switch as the best choice for most of them. This doesn't mean you can simply state that if a game has a million dollar prize and a one cent prize that it doesn't matter whether you switch, its more like the difference between saying:

Is it better to switch every time you have a million dollar prize and one cent prize? (1 case)

VS

Is it better to switch every time you know theres a million dollar prize left? (multiple cases)

For non-believers, I also quickly made an applet where you can test out scenarios. With this applet, you can either simulate a large number of games using some given conditions (basically locked prizes) or you can use interactive mode. So, if you would want to test out my theory on the top prize being locked: lock the top prize, put in 1000 games, make sure interactive is not selected, and click Next.

If you would like to see for yourself why this works, then try using interactive mode. This will essentially make it feel like a real game, but prevent you from picking a locked prize (since that is not the path we want to take).

You can also download the source.

## Thursday, May 29, 2008

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