lets make a deal article

Cecil's Columns/Staff Reports
1

i still dont see how it could be 2/3 odds to pick the other door, i mean you eliminated 1 door, the remaining doors are still 50/50 statistically, however in actual practice the other door is more likely to be the prize.

heres yet another analogy:

simple game, you shuffle a deck of cards and you have to guess the color of the card that is turned over. 3 reds are turned over, followed by 3 blacks. the odds are still 50/50 that it could be either, but in actual practice your better off betting on red since the last 3 were black.

another way is ok, your first choice has a 1 in 3 chance of winning, then one is eliminated. you got a 1 in 2 chance of winning, either.

if there were 5 doors you would have a 1 in 5 chance of winning, and 3 were eliminated, you would have a 1 in 2 chance of winning. thats it. theres only 2 doors the odds logically cant be less than 1/2.

and thats basically the point, if an empty door is always eliminated then theres only 2 options, both with logically equal odds.

2

i think basically what im saying is that the odds of winning when you switch are 2/3 because you have a 2/3 chance of picking the wrong one, and if the other wrong one is eliminated then the remaining door will have the prize, however logically its still 50/50. since one is right and one is wrong.

yes thats it.

3

[QUOTE=mattwhittall]
…simple game, you shuffle a deck of cards and you have to guess the color of the card that is turned over. 3 reds are turned over, followed by 3 blacks. the odds are still 50/50 that it could be either, but in actual practice your better off betting on red since the last 3 were black.
[/QUOTE]

Not quite. The odds are still 50/50 for either red or black.

4

The chance that you were wrong in the first place were 2/3. Monty opening an empty door doesn’t change anything, so the chance that you were wrong in the first place are still 2/3. Therefore, the chance that the only remaining door is right is also 2/3.

5

So, lessee, if I buy a lottery ticket, I have a 50/50 chance of winning, because either I win or I don’t?

6

Scooping the Dex-meister… Can I get Administrator as my title? Or maybe Ace Reporter?

When you start a thread about one of Cecil’s columns we like to include a link so that other people can read it and all know what we’re talking about.

Monty Hall/Let’s Make a Deal column

Oh, and welcome to the Straight Dope.

sinjin

7

Once more into the thick.

Let’s start with the obvious: You pick a door at random. You have one chance in three of picking the right door. You will randomly pick a losing door twice as often as you will pick a winning door. You accept this? Now let’s imagine the possibilities.

Possibility #1 - You picked the correct door. The other two doors are empty. Monty opens one of the empty doors and offers you a chance to pick the other one. If you switch you will lose.

Possibility #2 - You picked a wrong door. One of the other doors is empty and one of the other doors has the prize. Monty will open the other empty door and then offer to let you switch your door (which is empty) with the only door left (which has the prize). If you switch you win.

So we see the possibilities: If you pick the winning door and stay, you will win 100% of the time. If you pick a losing door and switch, you will win 100% of the time. So you should always stay when you pick a winning door and switch when you pick a losing door.

Now you obviously don’t know if you picked a winning door or a losing door. But go back now and reread what I wrote above: You will randomly pick a losing door twice as often as you will pick a winning door. Now reread the part where I wrote: If you pick a losing door and switch, you will win 100% of the time.

If you switch, you have twice as much chance of getting the prize than if you stayed.

8

ok you win.

but

say you were on a plank over a pool full of hungry sharks, and you were playing the red/black game. 3 reds were turned over, followed by 3 blacks. if you pick the wrong color, you go in the shark tank. would you pick red or black?

if your brave you would pick black, because the odds are the same right? me, i see 3 consecutive black cards i pick red. although, 2/3 of the time it is logically brown. but what would you do?

9

Matt, what you’re describing is called the Gambler’s Fallacy - the belief that past events in a random series will predict future events in the series. It has no validity and isn’t even consistent - some people believe that a series of recent reds means that red is “hot”; other people believe that a series of recent reds means black is “due”.

Consider this experiment to see if it convinces you. Shuffle a deck of cards and take the top six cards off the deck. Seperate the black cards from the red cards. Now stack one group on top of the other and put them back on top of the rest of the deck. Now turn the cards over one by one. Do you still think the color of the fourth, fifth, and sixth card has any effect on the color of the seventh card? If it does then you should be able to control the color of the seventh card by how you stack the first six cards.

10

It seems the Monty Hall problem has been discussed as recently as last month:

Here was my explanation from that thread:
let’s approach it from the car’s perspective.

The car is equally likley to be behind door 1, 2 or 3. (p=1/3 for each)
So there are only 3 possible situations if you pick (for example) door #1.

Situation A
Car behind door 1 - you switch and always loose. (Happens 1/3 of the time)

Situation B
Car behind door 2 - you switch and always win. (Car is behind an unchosen door and Monty can only pick door #3) (Happens 1/3 of the time).

Situation C
Car behind door 3 - you switch and always win. (see above) (Happens 1/3 of the time).

Adding up the probabilities, we see that by always switching, you win 2/3 of the time and loose only 1/3 of the time.

11

[QUOTE=mattwhittall]
say you were on a plank over a pool full of hungry sharks, and you were playing the red/black game. 3 reds were turned over, followed by 3 blacks. if you pick the wrong color, you go in the shark tank. would you pick red or black?

[/quote]

As Nemo says, it’s a mistake to look for “trends” in the way the cards are showing up. Or the dice, when you’re playing a dice game, or the balls in the lottery machine, or whatever.

In the simple card game you describe, the numbers of red and black cards still left in the deck are equal, and since the deck is shuffled, you have no basis for preferring either red or black. You are not “better off” picking either color. Pick the color that goes best with the shirt you’re currently wearing. Or flip a coin from your pocket. It won’t make a difference.

But if a thousand people play the Monty Hall game, and 500 decide to switch, and 500 to stay, then only 1/3 of the Stayers will win (because there’s only a 1/3 chance that anyone’s first pick is the prize door), and 2/3 of them will lose — whereas the numbers are inverted for the Switchers. Wherever a Switcher would win, a Stayer would lose, and vice-versa.

[QUOTE=mattwhittall]

if there were 5 doors you would have a 1 in 5 chance of winning, and 3 were eliminated . . .

[/quote]

This is an even clearer example, but I’m afraid you reached the wrong conclusion. In the 5-door Monty Hall game, the player picks a door, then Monty opens and eliminates 3 duds, then asks the player whether he’d like to switch from his chosen door to the one remaining. If a thousand people play this game, the Stayers will win 1/5 of the time, and lose 4/5. The Switchers will lose 1/5 of the time and win 4/5. Therefore it’s better to be a Switcher.

12

i conceded defeat already! in fact my second post pretty much proves myself wrong anyway. but i was at work and couldnt really think about it too much.

as for the red black game, i know its still a 50/50 chance, but there is something subjectively wrong with picking black given the circumstance.

if there were 1000 people in a row, all teetering over a vat, and they all picked red. around half would be shark food. but i would still pick red with confidence. why? how could something that feels so right be wrong?

13

[QUOTE=wolf_meister]
It seems the Monty Hall problem has been discussed as recently as last month:

Here was my explanation from that thread:
let’s approach it from the car’s perspective.

The car is equally likley to be behind door 1, 2 or 3. (p=1/3 for each)
So there are only 3 possible situations if you pick (for example) door #1.

Situation A
Car behind door 1 - you switch and always loose. (Happens 1/3 of the time)

Situation B
Car behind door 2 - you switch and always win. (Car is behind an unchosen door and Monty can only pick door #3) (Happens 1/3 of the time).

Situation C
Car behind door 3 - you switch and always win. (see above) (Happens 1/3 of the time).

Adding up the probabilities, we see that by always switching, you win 2/3 of the time and loose only 1/3 of the time.
[/QUOTE]

So why does situation A have the premise “you switch and always lose”?
Why couldn’t the premise be Situation A: Car behind door 1- you keep door #1 and win. (Happens 1/3 of the time).

Or are we saying that if door #1 contains the car, and I pick door #1, he will not open a door #2 or #3 with a goat?

14

[QUOTE=mattwhittall]
i conceded defeat already!

[/quote]

Sorry.

[QUOTE=mattwhittall]

as for the red black game, i know its still a 50/50 chance, but there is something subjectively wrong with picking black given the circumstance.

if there were 1000 people in a row, all teetering over a vat, and they all picked red. around half would be shark food. but i would still pick red with confidence. why? how could something that feels so right be wrong?

[/QUOTE]

Probably because the human mind is eager to find patterns, and will find them even when they don’t really exist. A run of three red cards followed by three blacks might look like it’s following some sort of rule. But, “R-R-R-B-B-B” is really just one sequence out of the 64 possible sequences for 6 cards, and all those sequences are equally likely.

In a favorite psychology experiment that I’ve read about, a bunch of rats and a bunch of humans interacted with a simple piece of apparatus: a pair of buttons and a pair of lights, one of each on the left, and one of each on the right. The subject, rat or human, would press one of the buttons. Then one of the lights would light up. If the subject had pressed the correct button — the one associated with the light that lit — he would be rewarded with a food pellet. Or probably money, for the humans. But I don’t know. Food pellets can be pretty yummy.

Anyway, the goal, from the subject’s point of view, was to predict over and over which light was going to light up next. The more of those you got right, the more you were rewarded. Of course at the start of the experiment, there would be no information to work with. But after say ten turns or so, you’d have a small stream of L’s and R’s to examine for patterns, that you could use to predict which button to hit next. After a hundred turns, you might get quite good at guessing what the machine was doing.

So which species was better at finding the patterns? Who won this little inter-mammalian battle of the brains? Sorry to say, it was the rats.

Because it turned out that there never was a pattern. The machine was simply lighting L’s 80% of the time, and R’s 20% of the time, in a completely random fashion. (Like rolling a five-sided die.) After a short time, the rats never bothered trying R, and simply hit L always, thus giving them a score very close to 80%.

Humans on the other hand kept looking for meaning in the madness, and collectively scored something much closer to 50 or 60%. Some test subjects would explain afterward the complicated rules they thought they had discovered. Of course there were no rules, and prediction was a hopeless exercise all along. The rats actually had the optimum strategy for this particular little game.

15

lol that is an interesting experiment indeed. your probably right about looking for patterns. they do make up a pretty large part of our intelligence tests.

what was the name of that book about the experimental super intelligent lab mice that escaped? i think they made an animated film about it as well.

anyway yea good stuff.

16

[QUOTE=mattwhittall]
what was the name of that book about the experimental super intelligent lab mice that escaped? i think they made an animated film about it as well.

[/QUOTE]

Mrs. Frisby and the Rats of NIMH was the name of the book (the one I’m pretty sure you’re thinking of at least). The movie version of it was called The Secret of NIMH.

17

Hampshire
Yes there were conditions from that other thread that I did not include in my “summary”.
The basic premise is that you will always switch after you are shown the goat.