I know, 50/50 heads tails right? But help me understand the next step – each coin flip has a 50/50 shot of heads or tails. What I don’t understand is how the likelihood of the next flip doesn’t change. For example if I flipped a coin 10 times and every time it flipped heads, the next flip would be 50/50 tails. Wouldn’t the likelihood of flipping a coin 11 times and having it be heads every time be really low? 0.5^11 = 0.048%?
Here’s the origin of the question. I was at a roulette table and the guy said “it’s been black the last 8 rolls, the next one has to be red.” At first I thought, the next roll will be ~47% black, ~47 red, ~6% 0 or 00 you fucking imbecile. Then I thought to myself, what are the chances that there are no red rolls in 9 rolls, which is well below 1%.
Am I the imbecile?
In: 215
The past does not influence the future.
The odds of a coin toss coming up heads is 50% and coming up tails is 50% for each time you toss it.
If you toss a coin 9 times and get 9 heads, that’s a pretty low probability (0.39%). But on the tenth toss, the odds for that toss are still 50% heads and 50% tails. The fates are not working to “even out the odds”. There’s no metaphysical push to undo unlikely events that have already happened and “balance things out”.
The problem is that people think “well the odds of the tossing 10 heads in a row are even lower (0.2%)”….which is true. But the unlikely series of events (the 9 coin tosses in a row coming up heads at 0.39% probability) has already happened. From that point forward, the odds of that tenth head is just 50%. But the odds of the ten heads in a row is 50% x 0.39% or 0.2%.
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I understand it’s contra-intuitive, but every single event has its chance independently from the previous ones.
Imagine you are tossing coins 8 times. One such run could be something like HHHTHTTH. You would probably call it a kind of realistic run. Even though the first 7 times you had slightly more heads (4H vs. 3T), it’s kind of okay to have yet another H.
Now imagine you exist in 256 parallel universes, and in the first universe you have a run such as HHHHHHHH. In the next one you have HHHHHHHT. Then HHHHHHTH, or HHHHHTHH, eventually HTHTHTHT etc. Of course there will be such “random” looking runs like HTTHTHTT, which is kinda the thing you expect.
Now those alteregos of yours where your coin did a “weird” run, they would think that this is impossible (or unlikely), and for example after 7 H’s there is more likely for a T to come. But in fact there will be only one universe where after 7 H’s in fact a T comes.
It’s because each exact run (even each regularly random looking run like HTTHTHTT) has only one chance out of 256. And each “weird” one has the same 1 in 256.
So when you are tossing the coin and have already 7 H’s, you can always imagine that you are one of your alteregos in 256 parallel universes, and one yourself will get a H, the other yourself will get a T. You just don’t know which one you really are, that’s why it still remains 50-50%
If this is not convincingly for you, you can always ask another question. If the coin (or roulette or whatever) has memory, how long is this memory?
Let’s say you have a coin and you already have 7 H’s. Let’s assume that for some reason a T has a higher chance now. For how long does this higher chance for T exist? Within an hour? Within a day? What if you never have the chance to toss the next time, but someone else does it for you? Is it okay to always have H’s, if you change the coin, because you “exhaust” the heads in one coin but have some left in the other? If you can exhaust a coin, what happens if you give it away? Can you cheat a coin toss (like the beginning of a football match for example) by exhausting a coin of heads and then tossing it because now it is surely tails?
If the odds are not linked to a coin but to a person (so you exhaust *yourself* of heads), what happens if two people meets and do a common coin toss; if one has had only heads previously and the other has had tails?
As you see if the luck had any memory it would create a super weird world.
Gambler’s fallacy: a random event is influenced by past results.
Truth: no matter how many heads or tails in a row occur, the probability of either occurring on the next toss remains 50/50.
What *does* change is the *liklihood* that such a runs of a particular length are seen over an infinite number of tosses.
So if you were to toss a coin 1,000,000 times, you would expect to record far more discrete instances of 4 heads occuring in a row than 40 heads occuring in a row.
Each toss has a 50/50 chance of going either way. But the frequency (and liklihood) that a sequence of all heads occurs decreases as the length of the sequence increases.
That guy was describing [the gambler’s fallacy,](https://en.wikipedia.org/wiki/Gambler’s_fallacy) and since he was gambling while describing a fallacy as a scientific fact, everything checks out
If you were to go by the logic that previous flips matter, why wouldn’t coin flips you made 1 year ago or coin flips made in another country not also matter?
There are many sequences you can flip a coin in and they all have equally likely chances. Tails or heads 3+ times in a row stand out more but we don’t bat an eye to mixed sequences like H T T or T H T, of which there are more than “perfect” ones. The likelihood of flipping heads 11 times in a row is small, but every other sequence is also unique, even though it seems more random.
BEFORE you start flipping, the odds of a run of 9 heads (say) are (1/2)^9. But.
If, at some point, you’ve already flipped 8 heads in a row, well – it’s now a fact. There’s no chance it didn’t happen. Its probability is 1. The chance NOW that you will complete a run of 9 is the chance of a run of 8 which has already happened, multiplied by the chance of another head. 1 x 1/2, i.e 50%.
The gambler’s fallacy is, basically, failing to recognise that the probability of something that has already happened is 100%. It doesn’t matter that what has already happened was, at one point, extremely improbable. It’s happened. It was improbable back then; now it’s not only not improbable, it’s a certainty. And the dice (etc.) don’t have memories, and don’t care.
The odds of any specific sequence is the same. 10 straight heads, ten straight tails, first five heads last five tails are all equiprobable.
Also, we assume the events are independent, the coin isn’t affected by the previous flips.
Here’s a perspective that clicks for me: After the roulette table lands on black 8 times in a row, the probability that it happened *is now 100%*, because it’s already happened. The chances of the next spin landing on black (or red) is 47%, as you say.
Every time something happens, you can think of its probability resolving to 100% since now it’s in the past and you know the outcome.
That doesn’t conflict with the idea that landing on black 8 times in a row is a very low chance (0.47^8). The trick here though, is to realize that *any* combination of 8 red/black outcomes also has this same 0.47^8 chance of happening. There’s nothing special about it being 8 blacks. A sequence of RBRRRBRB has the same tiny chance to occur, we just don’t focus on it because it’s not so nice of a pattern.
Imagine that instead of flipping a coin with “head” or “tails”, you flip 10 different types of coins. One is head/tails, another is black/white, another is sky/sea, another is summer/winter, etc.
You are about to flip your last coin. This coin is labeled, let’s say king/queen.
What are the odds you get either king or queen? 50/50, right?
Suddenly, a man bursts into the room. “WELL ACKTUALLY I labeled each side of your coins as “head/tails” in my head, and you got head all the time! So you’re guaranteed to get tails this time or it would be weird! Head is king, so you’re getting queen FOR SURE!”
Another breaks a window and yells “NONSENSE! I got the exact same idea and they all got head like you said, but in MY PAPER head is queen! You’re about to flip king!”
Keep in mind the coins you’re flipping only differ from the pattern on each side, and are otherwise perfectly balanced.
Who is right?
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