For example, when I flip a coin, I have a 1/2 chance of getting head, and the same chance at getting tails. With that theory, if I toss a coin 50 times, I should get 25 heads, and 25 tails.
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However, I did 3 rounds of 50 coin flips, with results of:
1. 28 heads, 22 tails
2. 27 heads, 23 tails
3. 27 heads, 23 tails.
I assumed that perhaps the coins weren’t “true”. Maybe the sides are waited different, maybe I tossed with different height/force each time. So I went virtual, and switched to having a computer roll a dice.
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I should have a 1/6 chance at rolling a number between 1-6. So 60 rolls, should have each number come up 10 times. But in practice my results were:
1. 14
2. 9
3. 8
4. 13
5. 6
6. 10
So how come practice/reality, doesn’t align with theory, even when we take human error out of the equation?
In: 34
There’s one thing called probability and there’s another thing, distribution.
Probability applies to the single event, like the coin toss. If you do several coin tosses (let’s say 6 of them), you will get different outcomes. You may get 3+3, but you may also get 4+2 or 5+1. You even have some little chance to get a straight run of 6 of tails or 6 of heads.
If you count all possible unique outcomes, there will be 64 different cases. I call it a unique outcome if the order of the heads and tails are different.
So for example HHHTTT is different from HTHTHT, however both of them gives you exactly 3 H’s and 3 T’s.
Out of the 64 possible cases, there’s exactly one that has 6 H’s (HHHHHH), and exactly one that gives you 6 T’s. You will also get 6 different cases of 1H5T (such as HTTTTT or TTHTTT), and another 6 of 5H1T. There will be another 15 cases of 2H4T and 15 of 4H2T. And the remaining 20 gives you 3H3T.
This thing is called *binomial distribution*. What it says is that even though the two outcomes have the same probability, only 20 of 64 cases will give you the expected 3+3 outcome. 15 of 64 will be 2H4T and so on.
Now imagine you have 64-sided “dice” (I’ll call it d64). This imaginary object has 64 sides and when you roll it, they all have equally possible probabilities to come. On each side you can have one unique outcome of a 6-coin toss.
So instead of tossing 6 coins you can roll a d64, and you can get, for example, HHTHTT or so. If you are only interested in the amount of H’s and T’s, then you will see that 20 of 64 sides of d64 give you 3+3. Basically what you did is you “packaged” the outcomes of the binomial distribution. It’s interesting to notice that although the 3+3 outcome is the most abundant, it’s only less than 1/3rd of all possibilities, so in most cases you will get one of the non-expected outcomes.
If you toss let’s say 24 coins, you expect 12+12, but because of the binomial distribution, 16-million sided dice to roll, and there will be a lot of non-expected outcomes such as 11+13 or 10+14.
If you go higher with the number of coin tosses, two things will happen:
One is that you will get results very close to the 50% expected outcome, and there will be virtually no huge deviations such as 30% and 70%.
Two is that the exactly 50% (like exactly 5000 of 10000 and not 5001) will be super rare, as this will be a very small portion of the all possible outcomes.
A lot of answers talking about how probability actually works but that’s honestly kinda irrelevant. Physics will accurately predict the outcome of any coin flip if you have all the necessary information to plug into the equations. Probability is just a way of coming up with a best guess when we don’t have all the necessary information.
But if you place the coin in a very particular way on your hand, flick it in a very particular way with a particular amount of energy, then the coin will follow a particular arc in the air and flip a particular number of times, hit the ground at a particular angle with a particular speed and then perform a particular series of bounces. This means that if you can flip a coin in a consistent way, then it will consistently land the same way too, skewing your results.
Using a computer to randomly generate results won’t guarantee different results every time either though. That’s because computers can’t actually generate truly random numbers. They generate what’s called pseudorandom numbers. There’s a very particular formula they use for generating these numbers that are based on a single starting value known as a seed. You can specify a seed manually, which will result in the exact same outputs every time. If you don’t specify a seed, then the computer will pick its own based on the current time. This gives an illusion of randomness since it’s never the same time twice, but the equations used to generate the numbers aren’t guaranteed an even distribution of the possible outcomes.
In short, nothing is ever actually truly random. Everything can be accurately predicted if you know enough of the variables. And probability is just a framework for making educated guesses when you don’t know all the variables.
Your sample size is small. 50-50 is the chance but it does not dictate the outcome. The more you flip the more you will trend towards 50-50. Think of it this way, if you decided to only flip a coin twice you wouldn’t be shocked to get H-H or T-T would you?
The best way to look at this is to break it down. Let us start with the computer one. Computers use something called “pseudo random number generation.” This is like asking a kid to randomly pick numbers. Sometimes he kid might try to make sure there is no pattern and other times the kid might say “5” every time. Think of that kid’s brain as what’s called a “seed.” Computers will use a seed as a basis of an algorithm (math equation) to generate a “random” number. You can use all kinds of things as seeds such as time, other equations, constant numbers, etc. Every seed will produce a type of pattern in the results. People can mitigate this with more complex algorithms and seed generation. However, for most “random” generators you find online, they will have a very simplistic generator that will be “biased” towards certain outputs as the computer can’t “really” pick a random number.
Now if we look at the coin toss we can break down “probability” and “statistical likelihood.” If you flip a coin and it lands on Heads (H) the next flip has (ideally) a 50/50 probability to be H/T and a 50/50 statistical likelihood of H/T. If you flip it 10 times, the probability that each individual flip is H is exactly 50%. However, the “likelihood” that EVERY flip results in heads is 1/1024 (possible outcomes ^ attempts = 2^10 = 1024). So we can approximate that if you flipped that coin in sets of 10 for ever and ever and never stopped, on average you would have a full 10 H set every 1024 sets of 10. That said, even if you flipped it 9 times and all 9 were H, there is a probability and likelihood of 50% that the 10th flip is also H but there is only a 1/1024 chance that all 9 flips before AND the 10th flip were/are H.
In other words, the likelihood of you getting hit by lightning once in your life are 1 in 15300. However, every thunderstorm you’re in you either do or do not get hit by lightning. 50/50 for the binary result of “did it happen” but the statistical likelihood is improbably small for “will every charged particle between you and the sky form the perfect discharge path for the lightning bolt to hit you.”
Statistical likelihood cares about every step of the process from the start to the end, and probability only cares about the next result as if it were the first result. If you flipped your coins until the end of time you would get almost an exact 50% H and 50% T distribution (assuming it was a perfect coin and perfectly repeatable flip).
> For example, when I flip a coin, I have a 1/2 chance of getting head, and the same chance at getting tails. With that theory, if I toss a coin 50 times, I should get 25 heads, and 25 tails.
The coin does not know that you plan on flipping it 50 times, and it does not know how you expect it to behave. If coins, dice, etc, behaved so that every time someone flipped or measured them some set of times, they gave exactly an even distribution of results, they would not be random, because they would be predictable.
People are really good at looking for patterns and expecting certain behaviors based on past observations. We are so good at it, that we see patterns in things that are inherently patternless and then don’t understand why those patternless systems don’t behave the way we expect them to.
the 50/50 coin toss really isn’t 50/50 — it’s closer to 51/49, biased toward whatever side was up when the coin was thrown into the air. Sorry I can’t give a ELI5 explanation, but I can atleast let you know your initial assumption of 50/50 is not entirely accurate.
https://phys.org/news/2009-10-tails-key-variables.html
For your rolls, the average is 3.3
For one of each on the dice it is 3.5
This is close enough to call it inside the statistical normal window
your assumptions are false, you shouldnt get exactly 50/50 heads and tails.
however you *should* get close to 50/50.
if you were forced to get 50/50 no matter what then the flip is rigged, like you would get heads and tails following each other everytime to never get away from 50/50 no matter the number of times you flip.
In order to answer this question, we need a better understanding of what the study of probability actually *is*. There is no magic force in the universe that is watching you flip coins and making sure that, at the end of the test, the results will equal 50/50. When you flip a single coin once, there are two possible outcomes, either heads or tails; it will not divide into half-heads and half-tails.
So what, exactly, does probability *mean*? Or to put it another way, what is it *useful* for? What is it actually *measuring*?
Probability only really *means anything* when you are dealing with a large number of tests. What it “promises” is that, the *more* tests you have, the *closer* the “real” values will align with the “expected” values. That’s what probability actually *is* – the distribution of results that the real values *approach* as the number of tests approaches infinity. The way it is typically measured is by performing a very large number of tests and examining the distribution of results.
Probability cannot tell you what will happen in any individual test, since any of the possible outcomes can happen. If you get a low-probability outcome, probability has not “failed” – you just happened to get one of the outcomes that we know must happen *sometimes*, just less frequently than the common outcome.
What it *can* tell you with a high degree of certainty is what will happen if you perform a large number of tests. It is therefore most useful in situations involving large numbers.
For example, people who make plans involving entire populations, such as politicians, economists, or epidemiologists often use statistics to predict the outcome of their decisions. They cannot predict the behavior of any one individual within the population, but if their numbers are accurate, the *overall* outcome on the entire population could be extremely predictable.
The probability of one flip resulting in heads is 50%. When you flip it again, the coin doesn’t know what it did on the previous flip, and the probability is again 50%. Hence sampling variability.
Expecting the theoretical probability over small sample sizes is essentially the gambler’s fallacy.
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