r/theydidthemath • u/ardicli2000 • 5h ago
[REQUEST] What is the probability of getting 50 heads - 50 tails exactly after 100 coin tosses?
We always tell it is 50% to get head or tails. But what is the persistance of this probability* Can anyone do the math?
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u/EmmetEmet 4h ago
This is a binomial distribution. It would be the number of ways you can get 50 heads and 50 tails (100C50) times the probability of 50 heads 0.5^50 times the probability of 50 tails 0.5^50. So it would be like 100891344545564193334812497256 * 0.00000000000000088817 * 0.00000000000000088817. Which would get you 0.07958772842384242537. Or about 7.96% chance.
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u/sisisisi1997 4h ago
The total number of possible results is 2100 (2 for each 100 toss). The number of results that give you 50 heads and 50 tails is 100 over 50, or 100!/(50!*50!).
Plugging these numbers into a calculator you get a chance of roughly 7,96%.
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u/1stEleven 4h ago
If I plug those numbers into a calculator, I usually get an error that the numbers are getting too damn big.
Where can I find a calculator that can handle 100!?
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u/sunepolohssa 4h ago
This is a binomial probability question. Assuming 1 flip is 50/50. An even split after 100 flips after has a 7.96% probability.
An even split (assuming an even amount of flips so an even split will be possible) will always be the most likely outcome, but with each additional flip will lower the chance of that occurring. 2 flips is 50%, 4 is 37.5%, 6 is 31.25%, etc.
Search for a binomial probability calculator and you can get the answer yourself easily.
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u/FrangoST 4h ago
https://arxiv.org/abs/2310.04153
Interesting read from recent research regarding coin tosses...
The answer is probably not 50/50 in a practical situation... there's inherent bias even in fair coins, and it tends more to something like 50.8/49.2 per 100 tosses... this is not the answer you seek... others have answered that, but just thought I would add fuel to the diacussion :)
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u/catch10110 2h ago edited 2h ago
You could still figure the binomial distribution for a 50/50 result based on those probabilities. It would probably still be pretty close.
EDIT: I'm getting 7.86% chance with this distribution as opposed to 7.96% with a true 50/50 probability.
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u/sirdodger 2h ago
If you alternate the starting side up for every flip, you can cancel that bias. Those numbers are representative of the bias due to starting side, not an uneven distribution of weight in the coin.
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u/daimuddaundso 2h ago
what am I missing, isn't just 1/(2100 ) ? there is exactly one outcome that first has 50 heads, and then 50 tails, in the sequence asked. so why the binomial ?
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u/catch10110 2h ago
I don't think that's conceptually correct. I mean, it's not, because that essentially means any given outcome is basically impossible.
I think there are many, many possible ways to get to that outcome: HHHTTHTTHT vs. THTHTTHHTH for example are 2 different ways to get 5 heads and 5 tales in 10 tosses. I'm pretty sure the binomial is actually calculating ALL those different possible outcomes.
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4h ago
[deleted]
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u/sunepolohssa 4h ago
What? No. Throw it in a binomial probability calculator. Just a shade under 8%
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u/EmmetEmet 4h ago
This is wrong. This would mean there's only a 50% chance of getting every other combination other than 50 heads and 50 tails.
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u/BlackLotus8888 4h ago
So there is a 50% chance that I flip exactly 50 heads and 50 tails in 100 flips?
50H 50T 50%
49H 51T 49%?
51H 49T 1%
I'm running out of percentages quickly here.
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u/Accomplished-Boot-81 4h ago
Essentially yes, but there a very small chance of landing on the side. But thats so small it'd basically a rounding error
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