Lecture Scene

Just curious, there was a lecture scene where Micky gave Ben 3 "doors" to choose and then Ben chooses one. After that, Micky gave away the result of one door and now gives Ben another choice to choose the "winning door." Ben decides to choose a new door instead of staying with his original decision. What was the math reason behind that?

Does it have something to do with how his original decision only gave him 1/3 chance of winning, but if he were to choose again, now he gets 1/2 chance to win whereas if he were to stay with his original decision he was keeping his 1/3 chance?

Thanks

This is the famous Let's Make a Deal Question discussed in most intro stats courses. The trick here is that the game show host knows which door the car is behind and therefore can and will always show a goat. If the game show host rolled dice to determine the door and just happened to show a goat then the probability would be 1/2, however that is not what he does.

The easiest way I can think to explain this as follows the origional probability of getting the car is 1/3. Because the host knows which door the car is behind, he can always show you a goat and does so. This in turn doesn't change the probability on your door. Since the probability that the car is behind your door is 1/3 and the sum of all probabilities must add up to 1 the probability that the car is behind the other door is 2/3 so it is beneficial to switch.

The fact is, if you choose door #1 then they eliminate door #3 how can there be a benefit to switching to #2. Both #1 and #2 have the same odds 50/50. There is just as much reason to switch as there is not to.

If you guys dont believe that switching wins more than try it out for yourself. You'll quickly see that switching wins much more than staying.

switching, etc

So, how does this relate to blackjack ?

golfnut101 said:

So, how does this relate to blackjack ?

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It doesn't. I was kinda suprised when he asked it. It had nothing to do with blackjack and nothing to do with non-linear equations, the name of the course.

non-linear equations

Non-linear equations ? Blackjack ? I've always enjoyed stats, odds, etc. but by no means am a math guy. Sounds like you are. How are the two related ?
Just a pure interest question.

thnx for your time

I'm not really sure I understand you. If I recall the scene correctly Kevin Spacey was talking about Newton's method and said the name of the course was non-linear equations, so basically they were doing calculus. Then he asked Bill Campbell a famous stats problem that really just involves tricky algebra and a knowledge of conditional probability. I didn't really see a direct connection between the two.

I wouldn't put too much thought into this. I think it's just another case of what you see in movies all the time; they talk about a subject that most people won't have much knowledge about and try to sound technical, but if you do happen to have specific knowledge on the subject it doesn't make sense. It's not important to the movie as a whole in terms of its plot or its entertainment value and the best method is just to accept it and move on with the plot.

The Wizard of Odds answered this

http://wizardofodds.com/askthewizard/122
and
http://wizardofodds.com/askthewizard/124

lol I couldn't help but laugh at this scene. The only reason to show this scene in a blackjack movie, is to show how "genius" Ben is. You don't have to be a genius to count cards but maybe you do to be amazing as the mit blackjack team was. However, they probably did not win so much because they were really smart but rather because they had a ton of money to start with.

The only reason one may choose the 2nd door, is because the gameshow host may want to mess with you and doing that could give away the answer. However, you would need to know the host's personality to know if he's the type who would do that. Your odds are 50/50 if you stay or leave.

A very prideful mathematician may explain this crap to an uneducated crowd or their class in order to show just how smart they are.

nck111488 said:

lol I couldn't help but laugh at this scene. The only reason to show this scene in a blackjack movie, is to show how "genius" Ben is. You don't have to be a genius to count cards but maybe you do to be amazing as the mit blackjack team was. However, they probably did not win so much because they were really smart but rather because they had a ton of money to start with.

The only reason one may choose the 2nd door, is because the gameshow host may want to mess with you and doing that could give away the answer. However, you would need to know the host's personality to know if he's the type who would do that. Your odds are 50/50 if you stay or leave.

A very prideful mathematician may explain this crap to an uneducated crowd or their class in order to show just how smart they are.

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its actually 2/3 chance if he were to change the door.

first of all thanks for the help

its hard to explain unless u have taken statistics in college, which fortunately, i have in the previous semester. i wouldn't have understood as easily without taken statistics.

so yeah. its more beneficial to change your decision you were expose the first door being the goat.

E-town-guy said:

The fact is, if you choose door #1 then they eliminate door #3 how can there be a benefit to switching to #2. Both #1 and #2 have the same odds 50/50. There is just as much reason to switch as there is not to.

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I thought about it last night and figured it out. People are right though that it doesn't fit into the movie plus they don't explain it very well for the avg person, probably because they want to sound more intelligent than they are.

E-town-guy said:

The fact is, if you choose door #1 then they eliminate door #3 how can there be a benefit to switching to #2. Both #1 and #2 have the same odds 50/50. There is just as much reason to switch as there is not to.

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Nope. When the host opens a goat-door, it's the same as the host saying "hey, if the car isn't behind YOUR door, it's behind this door." Since there's a 2/3 probability of the car not being behing your door, there's a 2/3 probability of the car being behind THE ONLY REMAIN DOOR. Really, just google "monte hall problem." You will find explanations longer than and superior to mine.

i think i get the math of the thing from looking at the wikipedia link but darned if it makes common sense to me. guess if it did so would quantum mechanics or maybe even women.

sagefr0g said:

i think i get the math of the thing from looking at the wikipedia link but darned if it makes common sense to me. guess if it did so would quantum mechanics or maybe even women.

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Definetly go with the Quantum Mechanics.! Much easier to understand.

godeem23 said:

Nope. When the host opens a goat-door, it's the same as the host saying "hey, if the car isn't behind YOUR door, it's behind this door." Since there's a 2/3 probability of the car not being behing your door, there's a 2/3 probability of the car being behind THE ONLY REMAIN DOOR. Really, just google "monte hall problem." You will find explanations longer than and superior to mine.

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Well, I never even made it through trigonometry, but even after having this explained, it still doesn't make sense to me. To me, it seems the odds are 50/50.

Regardless of which door you choose, when the host takes away one of the "goat doors", you have one door with a car, and one door with a goat. You are then given the choice of which door to take. This is 50/50.
Since a goat door is going to be removed regardless of your first choice, that third door doesn't even matter. It is removed from the equation. The choice is always between two doors. Not three.

I may be wrong, but this is still the only way my limited mind can see this.

Playloud said:

Well, I never even made it through trigonometry, ......

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yea trigonometry is one of the worst along with analytic geometry imho. once past that nonsense lol it's not so bad. i never really was able to get a sense of understanding of trigonometry even as badly as i wanted to and as hard as i tryed. just barely passed the course by treating it as a cookbook follow the recipe sort of thing. one of those rush, rush courses where proofs and explainations went out the window. but this problem with the car and goats seems the same. i can do the math like a recipe but don't come away understanding what the heck how can this be.

Playloud said:

I may be wrong, but this is still the only way my limited mind can see this.

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When I just think about it, it's obviously 50-50. When I read the explanation, it's obviously 2/3-1/3. You know when they use the words "paradox" and "counterintuitive" in the explanation it's going to be tough to wrap your mind around it!

Conditional Probability

The principle involved is computing a probability given a condition.

Example from blackjack:

You have a hand of 10-10 v dealer ace and cards left in shoe are 4 aces and 4 tens -

Player EV for standing on condition dealer has checked for blackjack and doesn't have it = 100%
Here dealer will always have ace in the hole and will always bust since it is known dealer doesn't have blackjack, so player always wins.

Player unconditional EV for standing where player loses to dealer blackjack = 0%
Here dealer has BJ 50% of the time and player loses and ace in the hole 50% of the time and will always bust and player wins.

Once it is given that a condition exists, the EV changes.

k_c