Someone explain this to me because I’m missing something.
If I flip a coin and get heads 20 times in a row, why is there only a 50% chance I’ll get tails on the next flip. Isn’t getting heads 20 times in a row fairly rare? How about 1,000 times? Isn’t it more likely that I’ll get tails after 1,000 head flips?
I seem to be missing some simple fundamental point on this.
Because the odds of you flipping heads 1,000 times in a row are different from the odds that the next one is going to be heads.
There’s always only a fifty-fifty chance of you flipping heads on any individual toss. Each toss is independent from the past toss, so it doesn’t affect its odds.
Quite frankly, though, if you were to flip heads 1,000 times I’d definitely bet heads on the next flip because something is rigging that coin.
Because there’s always a 50% chance that you’ll get tails on any single flip.
Nope. The 1,000 head flips have already happened, so they are not factored in when determining the probability of the next flip.
Yep. The probability of a sequence of events is determined by the product of the probabilities of the single events that make up the sequence, but there is no relationship in the opposite direction. The probability of a sequence has no effect on the probability of any particular event in the sequence.
Basically, it’s been pretty well covered already, but let me try one more explanation, with smaller numbers. And the way you phrase the question is very important.
If you ask the question, what are the odds of me flipping five heads in a row, the answer is 1 in 2^5, or 32. There are 32 possible outcomes, only one of which is five heads.
If you flip four heads in a row and ask the question, what are the odds the next flip is a head, well, there are only two possibilities: heads or tails. If you’re using a fair coin, the probability of each is fifty percent. The coins have no “memory.” There’s no logical reason why a flip after four heads would affect the chances of another head coming up (unless the coin is rigged.)
Make any sense?
The short answer is: The coin has no memory! Since there are two possibilities in each flip, each one has a 50% chance, no matter what happened before.
Now the long answer. You are confusing a priori with a posteriori probabilities. The a priori (before performing the experiment) probability of getting 21 heads in a row is 1/2[sup]21[/sup], approximately 5x10[sup]-7[/sup}, so low as to be almost impossible.
The probability of getting 21 heads in a row, given that you already had 20 heads is 1/2 (a posteriori).
Let me put it into the context which I read it.
I was reading a discussion about that British guy who sold all his possessions and gambled everything on one spin of the roulette wheel. He bet on Red and won.
Someone mentioned that he should probably waited for it to get black 3 or for times in a row and then bet on red. Someone said “ah yes, gamblers fallacy”. They said there was a 50/50 chance even after 1000 spins.
I missed those explanations before I posted.
I think it’s sinking in.
How about a little more layman explanation of that priori posteriori stuff? I think that’s screwing me up.
If you get 1000 head flips in a row, I think it’s time to closely examine the coin. 
The “gambler’s fallacy” is that the wheel must be be [color] this time because it’s been [other color] the previous [whatever] times. As others have said, the previous outcomes have no effect whatsoever on additional trials. If he waited until black came up 1 billion times in a row, the odds of the next spin being black are the same as always (in theory; In reality, that would suggest that there is something funny going on with the wheel, as Neurotik said).
Another way to look at this- lottery numbers. If you beat the nearly impossible odds and pick 6 correct numbers in a lottery, the odds of those * same 6 numbers* coming up in next week’s drawing are the same as any other combination (practically zero).
Mathematically, it’s simply that the odds of an event that has already occured ocurring is 100%. So, for flipping a coin heads 25 times in a row after you flipped 24 in a row is:
chance of flipping 24 in a row * chance of the 25th (next)
100% * 0.50
Let try and rephrase what the others said.
When you figure out the odds of something unlikely happening, like Red turning up 10 times in a row on the Roulette wheel, you are judging the overall result of a whole lot of small decisions. Pretend you are walking a path and every few steps you come to a fork in the road. You decide that Red means left, Black means right. Each decision (red or black (left or right)) does not affect the next decision (red or black (left or right)), but DOES affect the ultimate destination.
Another way to, perhaps, clarify is to say that there are not a specific number of REDs in the wheel. It is not like a bag full of an equal number of red and black marbles where when you remove a red marble there are now more blacks. There are infinite reds and infinite blacks. Having one red result means there are still infinite reds left.
I don’t know if that helps at all
Thanks all, I think I’ve got it. A bit of a strange concept to wrap my brain around at first, but the explanations make sense.
A priori probability is the probability of having an event without previous information. A posteriori means that you have already some information.
In the roulette example: if you were to bet beforehand that the roulette would give 4 blacks in a row, your odds would be 1/16. This is a priori probability.
If you know that the roulette gave already 3 blacks, you can bet with 1/2 certainty that you will have 4 blacks in a row. This is a posteriori probability and jk gave you already the formula.
Interestingly, if a coin came up heads 10 times in a row, about half the suckers would believe it was surely going to come up heads again (“it’s on a streak”), and half would believe it was going to come up tails (“it’s due”).
Of course, coin-flipping is a better game than roulette, where the odds are slightly less than 50%, since 0 and 00 are neither red or black.
Probability can get complicated, but the basics are very simple. To get the probability of any given event, divide 1 by the number of possibilities. If you flip a fair coin, it can either land heads or tails. Two options. Hence the probability of either is 1 divided by 2, or one half. Note that what has happened before simply isn’t part of the equation.
On a minor note, the phrase ‘The Gambler’s Fallacy’ seems to be used by different people to mean different things. The commonest meaning I’ve come across is “When he’s winning he thinks his luck will stay the same, when he’s losing he thinks it will change.” But all the different interpretations amount to the same thing… gambling is a mug’s game. Which it is.
Not directly related to the question but consider the following:
The combination of 10 Heads (HHHHHHHHHH) is just as likely as (HHTTHTTTTH). They are each one distinct pattern out of 2^10 possibilities. Just because all Heads or all Tails looks dramatic, from a probability standpoint it is no more or less likely than any specific random pattern.
What about the coin landing on edge? Or being grabbed out of the air by a passing parakeet?
From my experience, Gambler’s Fallacy occurs because the human mind is so conditioned to look for patterns; so much so sometimes that the gambler’s natural assumption is that there is someone or something out there (“Lady Luck”? g_d?) that is remembering and controlling the outcome of the game.
I realize the question has been answered, but I didn’t even realize there was a term for this. I first read about this fallacy in Poe’s The Mystery Of Marie Roget, so thought it would be cool to post his answer:
I bought a whole bunch of pennies once: 1,024 of them. I flipped every blessed one of them ten times in a row. One of them (as you might expect) came up heads all ten times. I kept that one for good luck.
One day, a friend wanted to flip for something. I said, “okay, but only if I get to take tails.” I took my penny out of my pocket and handed it to him. The fool! He flipped it, thinking that, since it looked, weighed, and felt just like every other penny he’d ever handled, that it was a 50/50 chance.
Another completely false story
Once, a friend of mine bought a whole bunch of pennies (512 of them) thinking he could flip them each nine times in a row and that one would certainly come up heads all nine times, and then he’d have a penny that was “primed” to come up tails. Turns out there was a light breeze that day, and he had two pennies come up all heads, nine times in a row. What luck!!!
I purchased them both from him for $0.10 each, knowing their value, and took them out to a party, where I decided to win some bar bets with them. The first one worked just like he said it would!!! I won a beer off some poor schmuck who didn’t realize my penny was “primed”. Then, when I tried the other one, it failed me – it came up heads. I owed the guy a beer.
I went back to my friend the next day demanding my ten cents back, and he said “What do you mean, you fool? You paid me ten cents each for pennies that had come up heads nine times in a row. You used one of them up, but now the other one has come up heads ten times in a row!! If I only had $0.20, I’d buy it back from you!”
I walked away, even happier with my purchases – I was down eighteen cents, even on beers, and I had the ultimate betting penny.