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
Note that in your second test you did get exactly 10. On average over the six attempts. The theory just says that it will _approach_ that distribution, given enough attempts. Not that you always will get that distribution in any attempt.
When you flipped the coin, did you start from the same position each time (ie heads / tails), measure the force and speed the coin was flipped at to make sure that’s conisistent etc? So many variables here. The small sample size isn’t enough to really show trends either.
Theoretical probability is the most likely scenario to happen not the one that will.
There can be probabilities of probabilities too.
>if I toss a coin 50 times, I should get 25 heads, and 25 tails
No you shouldn’t. You should get *close* to that, which is exactly what you got. Probability is about the *likelihood* of something occurring, not the *certainty* of something occurring. A probability of 1/2 doesn’t mean some event will or must occur exactly 50% of the time, it just means something close to a 50% is the most likely outcome, and that averaged over a large number of times, you’d expect the event to occur roughly 50% of the time.
You’re not thinking big enough.
Why did you choose the number 50 for the coin flips? Would you expect that if you flipped it 2 times, that it you’d always get exactly one heads and one tails? No, of course not. You need to flip it enough times for it to even out. 50 is not nearly enough.
10,000 flips is a better number, but you’re probably still not going to get exactly 5000 heads and 5000 tails. The more flips you do, the closer it gets to 50/50.
Try 1000 dice rolls and you see numbers that are closer to 1/6. Try 10,000 rolls and it should be even better. Try 100,000 or 1 million and you should get really nice numbers.
There’s always an error probability, which is the amount that it deviates from the exact probability number. This can be calculated through something called the “standard deviation”.
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Many answers here are saying you will get closer to 50/50 if you flip more coins, which is correct.* But perhaps one way to grasp this concept is by thinking about it the other way. What if you flipped *fewer* than 50 coins? In the extreme, you can flip a coin once, and it will be either heads or tails. There’s no magic in probability that will force it to somehow be 1/2 heads and 1/2 tails. That would make the rate of heads in your experiment *extremely* off what you were expecting no matter what.
Flipping 2 coins gives you some chance of having 50/50 with 1 heads and 1 tails, but it’s definitely not impossible to flip a heads (or tails) twice in a row. You surely did this many times during your experiment. Once again, there’s no magic in the coin that looks at the last flip and says “Ah, so that was a heads, which means I have to flip tails the next time”. So with just 2 flips, we also have a pretty good chance of being very far away from the probability, though it’s not as bad as with 1 flip.
As you keep adding flips, you end up with more and more “typical” outcomes and fewer and fewer extreme outcomes, so you move towards the theoretical probability. However, you’re never guaranteed to reach it. In some sense, the whole field of statistics is just about characterizing that process and being able to say precisely whether results like yours (because sometimes you only have 50 observations!) are unusual.
*In the sense of the average number of heads. If you flipped a coin a million times and got 500,010 heads, that’s a rate of 0.50001, so very close to 1/2, but the absolute deviation is 10 flips, much higher than in any of the experiments you did.
You have the theory wrong. Others talk about sample size but that’s not exactly what’s going wrong here.
Take the simplest example of a fair coin. The theory states you should expect a flip to be equally likely to be heads or to be tails.
So what should you expect if you flip one coin exactly? Surely not half heads half tails. It will be one or the other. The space is H, T – there’s 50% chance to be either
Now if you extend this to two coins, what do you expect? Well the two flips are independent, so again before each flip both cases are equally likely.
HH, HT, TH, TT – there’s 25% chance to be any of the four.
But now notice that we increased the number of cases that are exactly half-half (in one coin we had zero cases, with two coins we have two cases). As we have more and more flips we will tend towards more cases that are more balanced in the number of flips.
Consider this: if you flipped a coin exactly once, with a 50% chance of heads and a 50% chance of tails, would you expect to get half of heads and half of tails in that one flip? No, you would get either heads or tails, with an equal probability of each. This result, with your sample size of 1, would be either 100% heads or 100% tails. You can flip a coin once more to see if you end up with the other result to balance it out, but you might also end up with the same heads or tails result again. The previous flip does not force the next flip to balance out the odds, it has the same equal probability for each result. This is because the probability of the result just applies for the one, current coin flip or dice roll. It does not take past flips into account to influence the odds of your next flip.
When we track lots of coin flips or dice rolls, we find that the results tend to converge towards the underlying probability, but nothing forces the results to align once you have enough flips. The only point at which results are forced to converge is when you flip an infinite number of coins, which you are not going to do. Before that, the results will be close but are not guaranteed to be exact, and will tend to get closer as you flip more coins. You can somewhat see this with your coin flips: while at first the results were 100% on one side, you ended up with results of 56% and 54% weighted towards heads, which is closer to the underlying probability. As you flip more and more coins, that number will, with some wiggles, move closer and closer to 50%.
Your version of the theory is a little too strict. Theory actually says that if you flip a coin 50 times, you’ll **usually** get **about** 25 heads and 25 tails. The “usually” part is important. Theory doesn’t say you’ll get exactly 25 of each. The theory also says the more times you flip, the closer to an even split you’ll get. (at least at ELI5 level).
Try going to this site and choosing the 10,000 times option. Watch the “heads” percentage as the coins flip more and more. At first you might see an extreme split, like 70% heads, but as more and more flips happen, the percentage split will get closer and closer to 50/50. I’m up to 200 flips now, and the split is 45% heads and 55% tails. After 300 flips it’s 47%/53%.
https://flip-a-coin-tosser.com/10000-times-flipping/
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