# Gambler's Fallacy

Hendross
Posts:

**762**Critical Contributor
Just a friendly reminder that, you are never due/owed anything in independent trials.

The odds of flipping heads 21 times in a row is 1 in 2,097,152. The odds of the next flip being tails is 50%

Conversely, given 21 attempts, there is a 99.9999523162841796875% chance of getting a heads, the odds of the next flips being tails is still 50%

The odds of flipping heads 21 times in a row is 1 in 2,097,152. The odds of the next flip being tails is 50%

Conversely, given 21 attempts, there is a 99.9999523162841796875% chance of getting a heads, the odds of the next flips being tails is still 50%

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## Comments

130Tile Toppler130Tile TopplerThere are 3 doors. 2 have a goat behind, and 1 has a car.

You choose a door. Then, the host opens a different door and shows a goat.

You have the option to change your answer. Do you keep your original door? or switch to the 3rd door?

931Critical ContributorThis is a great one, indeed. It's fairly simple to explain on a very, very basic level: there are three possible iterations of the set (car/goat/goat, goat/car/goat, and goat/goat/car); although revealing a goat eliminates one of them (and therefore you'd think that the odds your door would have a car would be 50/50), it does NOT eliminate half of the "losing" iterations. There are still two potential outcomes that have a goat under door 1. Ergo, you have a 1/3 statistical chance of winning the car. Because you know a goat is under door 3, there is only one potential outcome that has a goat under door 2. Ergo, your odds of winning if you switch are 2/3.

This, however, is very, very, VERY difficult to prove mathematically, and the odds actually vary quite widely based on what conditions you assume. (The base scenario assumes that the host MUST pick a goat AFTER you select your door.)

4,663Chairperson of the Boards130Tile Toppler762Critical Contributor"Imagine that instead of 3 doors, there are 100. All of them have goats except one, which has the car. You choose a door, say, door number 23. At this point, Monty Hall opens all of the other doors except one and gives you the offer to switch to the other door. Would you switch? Now you may arrogantly think, “Well, maybe I actually picked the correct door on my first guess.” But what’s the probability that that happened? 1/100. There’s a 99% chance that the car isn’t behind the door that you picked. And if it’s not behind the door that you picked, it must be behind the last door that Monty left for you. In other words, Monty has helped you by leaving one door for you to switch to, that has a 99% chance of having the car behind it. So in this case, if you were to switch, you would have a 99% chance of winning the car."

47Coin one I knew - door one is new to me....

1251,157Chairperson of the Boards157Tile TopplerProblem is quite thorough, and explains which assumptions are necessary for the various conclusions to hold (blind host versus devious host, etc.)

https://en.m.wikipedia.org/wiki/Monty_Hall_problem

859Critical ContributorGood article. I love this bit from it:

"Pigeons repeatedly exposed to the problem show that they rapidly learn always to switch, unlike humans."

1,168Chairperson of the Boards2,495Chairperson of the BoardsBut I don't think people really understands the problem more easily with 100 doors, they just chose to change more because now in their mind it goes from 1/99 originally to 50/50 after the 98 doors are opened (because then there is just two doors, so then it is 50% for them), so it is better to change because now chances look higher on the second time (50% > 1%), but not because they understand that if they change they will have 99% chance of winning, which is the difficult part to grasp.

I always struggle with this problem, the 50/50 always come like the solution and I have to remember that Monty Hall will always open doors with goats, this is very important, always will ask you to switch (not just sometimes) and it is all part of the same problem, not two independent problems. But it is one of those super counterintuitive problems, even for Mathematicians (you just need to read the first batch of replies the author got when the paper was presented).

3Just Dropped InMyth busters tested this.... always switching won something like 80 percent of the time

6Just Dropped InAfter the door is opened to remove one of the losing options the choice of whether to move to a new door or not is now 50/50.

So the initial choice has a 1/3 chance in winning. That much is obvious.

The second choice has a 1/2 chance of winning if you change doors. Also obvious.

However, based on the new choice of switching or staying you now have a 1/2 chance of winning no matter which option you choose.

I'm sure I'll be flamed to death for this but I'm right if you think about it.

4,449Chairperson of the Boardsawareof where the goats are, and whether or not you selected the non-goat prize. Two-thirds of the time, you selected a goat, so the host only has one other goat to show you and you can now switch to the not-goat using the reversed odds of your original choice. However, one-third of the time you have selected not-goat, so it doesn't matter which prize the host reveals, because you're going to see a goat either way.It's that simple: the host's choice is fully informed and not random*, and inverts your original guessing odds

onlyif you switch.On the other hand, if you really want a goat, hold on to your original choice. Goats are pretty rad.

* Dayv's pedantic corner: ok, sure, the host's selection of a goat could be random when the host has two unselected goats to choose from, but really he's always going to pick the cuter one. The host gets to keep that goat.

5Well said, DayvBang, especially the emphasized words.

For those that still don't get it, here's a quick YouTube video: Monty Hall Problem for Dummies.

6Just Dropped InAgain. Original choice. You'll be right 1 out of 3 times. Obvious.

Now one wrong choice is removed.

Second choice. Key word here is choice. Either switch door or stay.

Choose to switch equals 50/50

Choose not to switch is now also 50/50.

Your odds ONLY increase from the original choice if there is NO second choice. As in, after you select the original door the host will remove one wrong answer ONLY if you agreed to switch without being given that second choice.

392Mover and ShakerRegardless of what you choose to do, the car is behind the remaining door 2/3 of the time. You should probably switch if you want to win it.

47Just Dropped InYou pick Door 1, then Monty reveals a goat:

Switch 67% Stay 33%

You pick Door 2, then Monty reveals a goat:

Switch 67% Stay 33%

You pick Door 3, then Monty reveals a goat:

Switch 67% Stay 33%