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 chance of flipping 11 heads in a row is 0.048%.
The chance of flipping 10 heads in a row, and then a tail, is also 0.048%.
Events that have already occurred don’t have a probability. They already did or didn’t happen. It’s like saying what are the chances a guy named dragonbank would be born in his current state, move to all the places he moved to, work all the jobs he worked, study for a PhD in economics, and make a reddit account. Well you can assume the odds of each of those events are super rare. But we know they already have happened so simply adding on “and then answer my eli5 question” doesn’t somehow make the odds of the whole thing rarer. The rest has already occurred. Now the issue is figuring the probability of me answering your eli5 question.
From the standpoint of statistical probability, yes, every coin flip is an independent 50/50 event, and the likelihood of 11 heads in a row is 1/2048 or .048%. This means that for any given flip, you have no reason to expect one outcome more than the other; but if you record the outcomes of 2048 sets of 11 coin flips, there is reason to expect that nothing but heads will show up just once.
The probability of 8 fair American roulette rolls in a row coming up all red is (0.474)^8, or about 1/392. So yes, it’s not going to happen often, but the person running the table can safely expect to see it happen every now and then.
More importantly, though, keep in mind that it’s this person’s job to discourage players from feeling as though these are independent events as much as possible. If the previous eight rolls hadn’t been all red, the person running the table would have described the situation in terms of whatever the most unlikely recent sequence had been. Maybe the last six came up even numbers; or maybe the last five were all in the third column. So no matter what happens, there’s almost always going to be a way to describe the current sequence as uncommon.
Probability doesn’t have the capacity to tell you what is going to happen next. This is always true, even when you toss a coin repeatedly.
Probability tells us that at some point in the future, after a significantly large number of tosses of a perfectly balanced and impartial coin, you will eventually have tossed an equal amount of heads and tails.
This is mainly because of how probability works — it is descriptive of the past, it isn’t technically predictive of the future. It can never tell you what is actually going to happen to a single event, or a single coin. It approximates the aggregate events, and does it reasonably well with large numbers but poorly with small numbers. We don’t know what’s going to come up next. We don’t know how many tosses it will take to achieve the perfect distribution either. However, given enough assumptions and some abstraction, we use the probability predictively for practical things.
This is usually just fine for whatever we are doing, including tossing a coin, but it is never predictive of the outcome of the coin toss in the strict sense.
The odds of getting 10 tails then 1 head are the same as getting 10 tails and then another tail. Both series are equally unlikely (but ofc possible). So the odds of that last event are 50/50.
Each flip, you have 50% of getting heads and 50% chance of getting tails. That is true.
The odds of achieving the sequence of H-H-H-H-H-H-H-H-H-H-H are the same as the odds of H-H-H-H-H-H-H-H-H-H-T.
It is not a 50% chance, though. As there are many other sequences possible, such as T-H-T-H-T-T-T-T-H-T-H. And if your calculation is correct (I didn’t check), then every possible sequence is .048% chance of happening.
Now, if you haven’t flipped any coin yet, it is a .048% chance of getting all heads in 11 flips. However, if you ALREADY landed on heads 10 times in a row, all those sequences that DIDNT start with H-H-H-H-H-H-H-H-H-H (10 heads) are no longer a factor and your odds of getting 11 heads in a row are 50% AT THAT POINT.
In fact, each time you land on heads, your odds are no longer .048%; it rises with each flip.
TLDR: You start with a .048% chance of getting 11 heads in a row. And with every successful flip, your odds increase. Eventually, it becomes a 50% chance when you’re on your final flip.
Edit: So, with the roulette, you are right. B-B-B-B-B-B-B-B-B has the same odds as B-B-B-B-B-B-B-B-R. The other guy is getting confused with the thought that getting 9 blacks in a row is a low chance IF he predicted that prior to any of the roulette rounds. And even then, guessing which one will be red is pointless.
Getting head on the 11th toss after getting heads in all previous 10 tosses is not same as getting heads 11 times in a row. The former experiment has already established that 10 heads in row have been observed(which in itself is an astronomically rare event) and getting head in the next toss is always 50-50. If you consider the chances of getting 10 heads in a row in the first place, then you get the same chances as the latter case.
The coin doesn’t have a memory of how it landed last time. So whatever the recenthistory was, it still has a 50% chance of landing on either side.
It is unlikely it will land on heads 9 times in a row, so if you predict 9 heads in a row, before any tosses are made, you will pretty certainly be wrong. But if you do by chance get 8 heads in a row, and then predict heads on the next toss, you have a 50% chance of being right, because the coin doesn’t know that it already landed on heads 8 times.
Depends. Is your coin totally balanced, meaning it still has exactly 1/2 chance on falling either way (and no chance of standing on its edge ?). Then flips are independent from each other, and each flip **still** has the exact same chance to fall on either side. Even it it has previously fallen 100 times consecutively on “heads”.
Now, that’s *only* if the coin is really perfectly balanced, and the toss isn’t rigged (like, by having magnetized the coin, and put a magnet to ensure it always falls on the “good” side. Or calculating the required force, and applying it perfectly, in a way that would physically make it certain that it would fall the expected way.
A series of tosses, supposed to be independent, but showing 100% of result A, 0% of result B, is **possible**, even with a perfectly balanced coin, but diminishes exponentially with extra tosses. With 10 tosses, all “heads” has 1/1024 chance of happening. 16 tosses, all “heads”: 1/65536. The formula is 1/2^x (2 to the power of x), with x being the number of tosses.
So, yeah, it’s very unlikely, but not impossible.
The odds of getting 10 heads in a row is the same as the odds of getting any other combination of flips. Consider that the first flip is as important as the last to get 10 in a row. And all the others. AND the odds of getting any other combination – 10 tails, or 5 heads and 5 tails, or alternating heads and tails, or 3 heads, then a tails, then 2 heads, then 4 tails – whatever – are all the same as the odds of getting 10 heads. You just don’t notice most of the other combinations because they don’t look significant. Same with something like playing the lottery with the numbers of your birthday – there’s absolutely nothing special about your birthday except to you, and the odds are the same as with any other numbers. You just notice the pattern so you think it’s rare, but the odds are the same as all the other patterns.
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