Question about the movie "21"?

I haven't seen the movie, but I think you're asking about the "Monty Hall Problem"

This is named after the host of a game show called "Let's Make a Deal" which was popular in the 1960s and 1970s. I kind of barely remember it; I think audience members dressed up in weird costumes so they would be chosen by the host, Monte Hall. It wasn't as good a game show as "The Price Is Right," in my opinion, but it was just as cheesy.

Anyway, the contestant would be faced with three "doors" or "curtains." Behind two of the curtains, there would be a worthless prize, represented by a goat or cow or something. (Livestock isn't actually worthless, but nobody wanted a goat in the 1960s and 1970s because they were all very up-to-date, and goats and cows offended their modern sensibilities.)

But, behind one of the curtains was the real prize. Usually a dishwasher or something, but if the contestant was lucky, it was "A NEW CAR!" and one of those big 1970s-era Chevrolets that looks like a mattress on wheels would be spinning around on a big turntable.

If you were chosen as a contestant, Monte Hall would ask you to choose Curtain No. 1, Curtain No. 2 or Curtain No. 3.

So, say you're living, right now, in 1971, and you're the lucky contestant. Monte Hall asks you to choose 1, 2 or 3.

After you make your choice, Monte opens one of the curtains that you didn't choose, and reveals a worthless prize behind it.

Then, he asks you if you want to change your choice to the other remaining curtain.

(It's important to remember that Monte Hall gets his paycheck whether you win or lose, so he doesn't have any "strategies" to change based on what your choice is--if there's enough time in the broadcast, he always gives contestants this opportunity, whether they've made the right choice or not.)

So, the problem is basically just this: Should you switch, or should you stay with your original choice?

Like many exercises in conditional probability, this one has a surprising result. Intuitively, we suspect that it doesn't matter whether you change your choice or not, since the prize is either behind there or it isn't, and how could Monte's act of opening the "wrong" curtain change the probability of the prize location?

What this doesn't take into account, however, is that, by revealing the worthless prize, Monte has given you more information than you had when you made your original choice.

Say the valuable prize is behind 1.

Now there are three equally likely scenarios: You pick 1, 2 or 3.

If you pick 1 (the prize), Monte reveals either of the other two curtains. Both of them have a goat.

Then, if you stick with 1, you win. If you switch to the remaining curtain, you lose.

If you pick 2 (a goat), Monte reveals the goat behind 3.

Then, If you stick with 2, you lose. If you decide to switch to 1, you win.

If you pick 3 (goat), Monte reveals the goat behind 2.

Then, if you stick with 3, you lose. If you switch to 1, you win.

So, if you switch, 2 of these 3 equally likely scenarios result in a win.

On the other hand, if you decide not to switch, 1 out of 3 of these equally likely scenarios will result in a win.

So the answer is that switching has a 2/3 chance of winning, and not switching has a 1/3 chance of winning.

Notice also that you're twice as likely to win if you switch, since 2/3 is twice 1/3.

Oh, man, this problem hurts the head. "Common sense" says that the odds are 50/50 for either of the two doors. That's certainly what I thought when I saw the movie. But here is why I was wrong - what the "host" does is not random - i.e., he ALWAYS shows a "goat" and he NEVER shows you the door you picked. Since what the host does is not random, use a more extreme example to make it more clear. What if there were 10 doors? You would pick one and be 10% likely to be right. By "rule" the host would then open up 8 "goats", none of which you picked. Now you have the choice to stay with what you picked (which you had a 10% chance of getting right originally - and is still only 10% likely) or move to the remaining door - which is now 90% likely since the host helped you out by opening so many losers. As odd as this seems, it is because we know the host never shows you your door and always shows goats. The non-random acts of the host change the odds.

There are 3 doors. 1 contains the car and the others contain a goat. Teacher reveals that door 3 has a goat. Then Ben switches his answer from door 1 to door 2. Why did he do that?

It's called variable change. It's all about statistics.

When Ben picked door number 1, he had a 33.3% chance of picking the door with the car. Meaning he had a 66.6% chance of picking the wrong door(The goats). Then he was told a goat was in door 3.

Now lets put the question in this way.

If you had two choices, door 1 OR door 2 and 3 combined, what would you pick?

Door 1 = 33.3%

Door 2 and 3=66.6%

Obviously you would pick the second choice.(Door 2+3)

That's why he changed his answer from door 1 to door 2.

Planes Trains And Automobiles When John Candy is singing The Mess Around in the car and eventually ends up going the wrong way down the interstate.