posted
This topic is for anyone who has seen the show Numb3rs Friday, 10PM ET/PT You can use it for general discussion or whatever.

I have a question from last Friday’s episode that I haven't been able to solve. I don't know how to explain it but, why in "Charlie's Lecture" does changing the students vote increase her odds? Charlie said it would give her a 2/3 chance of winning, verses the 50:50 that I would have guessed.
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posted
I didn't see it, since I don't watch TV on Friday nights. But if you care to elaborate, I might be able to figure out the statistics.
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posted
Ok, the guy (Charlie) has 3 cards. One with a CAR on the back Two with a GOAT on the back

First: He asked the student to choose a card (using intuition to pick the CAR) She chooses the middle one. Second: He reveals the far right (a goat). Third: He asks the class if changing her guess will improve her chances. They all say it will just be 50:50. Forth: Charlie somehow says that changing her guess will make her odds 2/3. And reveals the far left card. (the CAR)
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posted
I watched it once, and it seemed fairly plausible, if a little melodramatic. I'm always a little leery of shows where the villian and the crime are tailor-made for the abilities of the heroes. It's a favorite trick of Saturday morning cartoons.
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posted
I enjoy it, of course I've always been the math nerd in my group. I wonder, has OSC reviewed Numb3rs?
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posted
Here's a good link for examining the three doors problem (which is a classic stats problem).

The key thing is that the person who removes the first choice (in this case, it sounds like he "forced" a choice, in the magician's sense of the word) knows what's behind the cars/doors.
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OK, all that matters is the total number of each possibility at the beginning! What is left has no bearing on what the probabilities are at the start.
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posted
Hey I "sort of" have a cameo in the latest show.

No, I personally do not appear anywhere in the show, but a copy of a book that I typeset is on the table in front of Charlie during the chess scene at the end of the latest episode.
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posted
Ack! This problem drove me nuts when I read it in the "Ask Marilyn" column, and her solution still seems wrong to me. Looking at the site that rivka linked to, I think that this person's point of view sounds the most logical.

quote: I've been reading the Monty Hall problem and would like to post the answer to demonstrate where most people math is incorrect.

Given 2 doors, one with a prize and one without, which should you pick?

I think you'd agree that odds are 50-50 you'll be right. The fact that I may have had 3 doors initially and opened one, or 1000 doors and opened 998, does not change the current problem. There is still only a 50/50 chance of winning.

Because the probability of the problem can be expressed in three different ways.

P1 - 1/3 chance prize is door #1, 2/3 chance it is not P2 - 1/3 chance prize is door #2, 2/3 chance it is not P3 - 1/3 chance prize is door #3, 2/3 chance it is not

Most of the logic I've seen has the following flaw. As soon as the contestant picks door #1, they immediately assume that P2 and P3 are not valid. Well, they are. So they follow only P1. Regardless of which door I pick, all 3 are still valid. It is only when Monty opens a door that things change. By opening door #3 we now have

P3 - 0/0 chance prize is door #3, 3/3 chance it is not.

Now here everyone says, "Hey by P1, then the 2/3 chance all goes to door #2' But then why not, "By P2, then the 2/3 chance all goes to door #1"? Both are incorrect. When Monty picks #3, the whole problem is changed to:

P1 - 1/2 chance prize is door #1, 1/2 chance it is not P2 - 1/2 chance prize is door #2, 1/2 chance it is not P3 - 0 chance prize is door #3, 1 chance it is not

Look at it another way.

What's being said is that if I pick #1 and monty shows #3, then 2/3 of the time #2 will win. So If I had picked #2 to start with and monty opens #3, then the odds go 2/3 to #1. Why would they change? What if I don't have to tell monty? What if I could write it on a secret ballot?

posted
That is true IF AND ONLY IF Monty picks a door at random. But of course he does not. Assume you did not choose the correct door initially -- what are the odds the door Monty opens will be the prize door?

Absolutely ZERO. Because he knows which one that is and does not open it. Therefore, the odds are better for you to switch to the one Monty chose NOT to open.
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posted
A better way of phrasing it: The door you initially picked still has 1:3 odds -- it has not changed. But the door that Monty chose NOT to open has 2:3 odds. Simply because he deliberately did not open that one.

As one explanation I saw put it, if you had the choice to stick with your original door or BOTH the other doors (with Monty not opening anything), which would you choose?
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posted
Amancer, What rivka said. The problem with the logic of the guy you posted is that he's treating the two choices (which door and then to swap or not) as independent when they're not. Instead, the state of the problem in the second case is dependent on the initial choice.

Think of it this way. Say you're given the same set up with the first choice between the cards being the same but, on the second step, instead of removing a card and asking you whether you want to switch or not, Monty asks you whether you want to bet that the card you chose has the prize or not. In this case, it's obvious that saying that your odds are twice as good when you bet that it isn't, because two out of the three times you're going to choose a card without the prize.

The presented problem is actually equivilent to what I just said. The logical error with what you posted is that Terry is assuming that, because a card is removed, now it's just a two state problem (in one state you chose the right card and in the other you chose the wrong one) when it's actually a three state problem (first state chose the right card, second state chose wrong card #1, third state chose wrong card #2). You're not actually chosing between two cards; instead, you're, like in what I said above, betting whether or not you initially chose the right card and there were two chances out of three that you didn't.

And, one of the wonderful things about statistics is that you can play this out and you'll find that the odds really are twice as good for switching instead of being equally likely.
Posts: 10177 | Registered: Apr 2001
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posted
I like this explanation, from one of the links above:

quote:What if there were 1,000 doors? You would have a 1/1,000 chance of picking the correct door. If Monty opens 998 doors, all of them with goats behind them, the door that you chose first will still have a 1/1,000 chance of being the one that conceals the car, but the other remaining door will have a 999/1,000 probability of being the door that is concealing the car. Here switching sounds like a pretty good idea.

posted
The fact is that he is removing a bad chance, a chance to "get a goat" so to speak, not a random door.

Lets set this up, and I will bet you 10G that I am right....and it will be proved to be right over and over again, given a large enough set of examples.

If you didn't see what was behind the door he showed, and he simply gave you the choice of having the one door you chose, or both of the other doors, then it would be obvious that the chances switching sets of doors (from the one door you have to the two doors you don't) would increase you chances of winning to 2/3's, right?

In effect, that is what he does. He just is telling you which door out of the two not to choose if you do switch. Those two doors, even the one you KNOW isn't the prize, still have a 2/3 chance of holding the prize, while the one you chose originally still only has a 1/3 chance.

Since the second subdivision (eliminating a door) occurs within a subset that doesn't include your original choice....since he eliminates 1 of 2 bad choices, but only from the doors you haven't chosen....the chance of the right door being in the set other than you choice is still 2/3, he has just reduced the wrong choice from that subset.

In other words, chance has no memory, and as Monty's selection isn't random, that logic is wrong....so while it seems logical that it should be a 50/50 proposition, that is not true.

We assume that since there are only two doors left the odds of choosing right are 50/50, but that is wrong because it doesn't take into consideration the number of original selections. The fact is that there is only a 1/3 chance of selecting the right door in the first place, so eliminating a wrong choice doesn't even the odds up to 50/50...it is still a 1/3 chance per door, so the advantage lies with the group with the most doors in it, that being the doors you have not selected.

quote:Originally posted by advice for robots: I'm always a little leery of shows where the villian and the crime are tailor-made for the abilities of the heroes. It's a favorite trick of Saturday morning cartoons.

afr, I agree. It's the fatal flaw of Numb3rs, and makes it seem very contrived and fictional. Sure, math or a professional mathematician could help solve various criminal investigations. But that many of them?
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posted
I'm sorry Morbo and Advice for Robots that you don't see the point of the show.

Of course Charlie--the professor-- could not solve that many that many crimes. But it demonstrates how math can be used in real life. And often feels good when I know everything their talking about.

Except Hall's problem, until now

quote: Originally posted by rivka:

As one explanation I saw put it, if you had the choice to stick with your original door or BOTH the other doors (with Monty not opening anything), which would you choose?

posted
Griffin, I do enjoy the show, at least some episodes. And I think most of the plots would work in a one-episode format like a movie.

But it's a little unbelieveable to see Charlie and his brother cracking crimes with math week after week.

Also, Charlie seems too well-versed in very different fields of math, instead of specializing like mathematicians do.
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posted
It's not as if Charlie has ever (correctly) used anything that wouldn't be recognized from a highschool "survey of mathematics" course.
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quote:Originally posted by Morbo: Griffin, I do enjoy the show, at least some episodes. And I think most of the plots would work in a one-episode format like a movie.

But it's a little unbelieveable to see Charlie and his brother cracking crimes with math week after week.

Also, Charlie seems too well-versed in very different fields of math, instead of specializing like mathematicians do.

You actually think TV shows should be believable? My goodness it is a TV show. I figure any TV show that makes science and math look good is a good show. Unless you'd prefer more tripe like Desperate Housewives on the air?
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posted
I don't know who ghost-bumped this thread. I referred to it in another thread. As for the thread necromancy, I dredged up "good, bad, Yo-mama" but I think that's it. Oh, wait, I did CT's tumnus mayfly as well.

quote:You actually think TV shows should be believable? My goodness it is a TV show. I figure any TV show that makes science and math look good is a good show.

I don't think that shows which portray science and math in a positive but false light are doing anybody any favors in that regard.

quote:I don't feel Numbers portrays math and science in a false light.

Do you think that the way it uses math is believable and accurate? That's what I'm referring to when I said "false".
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posted
Yes. While they condense it into an action packed time slot, all of the math on the program has been extensively checked for accuracy, and the illustrations that they use when the math is being explained to the lay person are all based on published research.

posted
I realized I also was responsible for bumping an old thread about Halo 3 that caused Breyerchic to almost have a heart attack or something.
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