probability question | Mathematics | Page 1

raven goblet Dec 2, 2022, 1:09 AM

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is there an equation for how close to 50% the number of heads will be as you increase the number of coin flips? i know it would get closer forever but i want an equation for it

languid kilnBOT Dec 2, 2022, 1:09 AM

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haughty belfry Dec 2, 2022, 1:30 AM

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what do you mean specifically by "how close"?

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because obviously some cases will be 0% and some will be 100%

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thus you need to refer to a certain range

raven goblet Dec 2, 2022, 1:32 AM

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like shouldn’t the standard deviation for how close it is to 50% over time consistently get smaller

haughty belfry Dec 2, 2022, 1:33 AM

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yea

raven goblet Dec 2, 2022, 1:33 AM

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haughty belfry thus you need to refer to a certain range

yeah i want an infinite range equation if possible

haughty belfry Dec 2, 2022, 1:33 AM

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oh i think you misunderstand

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let's suppose we run 100 trials where we flip a given coin 100 times

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on average 50 of the coins will land heads

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this is clear

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however maybe in around 50% of the cases it will be less than 48 or greater than 52

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in maybe 90% of the cases it will lie within 35 and 65 heads

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and in 99% of the cases it might lie within 20 and 80

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alternatively you can ask "what's the average deviation from 50 heads" which is also not so hard to calculate

raven goblet Dec 2, 2022, 1:37 AM

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yeah i’m looking more to like 1 million flips to 500 million flips would there be an equation for its deviation from 50% as the flips increase

haughty belfry Dec 2, 2022, 1:37 AM

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"deviation" is an imprecise word

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do you want the standard deviation

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or the average deviation

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standard deviation is more useful if you wanna do more sophisticated analysis

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average deviation tells you on average how far it is from 50

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y'know what ill just send both

raven goblet Dec 2, 2022, 1:40 AM

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thanks lol i’m not sure what i’m looking for really

haughty belfry Dec 2, 2022, 1:46 AM

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do you know what a probability density function is

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also do you want the more accurate version or the less complicated approximation (it's only less accurate for small numbers of coin flips)

raven goblet Dec 2, 2022, 2:06 AM

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more accurate would be better, im trying to use it for something practical

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sorry for late response

haughty belfry Dec 2, 2022, 2:07 AM

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ok

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I'm gonna write choose as $C(n, k)=\frac{n!}{k!(n-k)!}$ for simplicity of latex

unkempt questBOT Dec 2, 2022, 2:07 AM

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marival

haughty belfry Dec 2, 2022, 2:08 AM

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Assume that you flip a coin n times.

raven goblet Dec 2, 2022, 2:08 AM

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n is rolls k is possible outcomes?

haughty belfry Dec 2, 2022, 2:08 AM

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n choose k is used in the formulas

haughty belfry Dec 2, 2022, 2:08 AM

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haughty belfry Assume that you flip a coin n times.

The standard deviation is given as follows (give me a sec to type up)

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$\sigma = \sqrt{\int^n_0 \left(C(n, x)\frac{1}{2^n}\right)\left(x-\frac{n}{2}\right)^2 dx}$

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The mean absolute deviation is given as follows

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$MAD = \int^n_0 \left(C(n, x)\frac{1}{2^n}\right)\left| x-\frac{n}{2}\right| dx$

raven goblet Dec 2, 2022, 2:12 AM

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thank you, i’m still a little confused on what k is tho

haughty belfry Dec 2, 2022, 2:12 AM

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oh just a typo

unkempt questBOT Dec 2, 2022, 2:12 AM

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marival

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marival

haughty belfry Dec 2, 2022, 2:13 AM

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if you wanna do further analysis use the standard deviation

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if you just want the "average deviation" use the mean absolute deviation

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and if you are so inclined to learn further, i found this thread upon a whim one time

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https://twitter.com/10kdiver/status/1393636244731482112

10-K Diver (@10kdiver)

1/

Get a cup of coffee.

In this thread, I'll walk you through 2 probability concepts: Standard Deviation (SD) and Mean Absolute Deviation (MAD).

This will give you insight into Fat Tails -- which are super useful in investing and in many other fields.

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raven goblet Dec 2, 2022, 3:00 AM

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thanks mate

mighty reef Dec 2, 2022, 6:03 PM

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raven goblet is there an equation for how close to 50% the number of heads will be as you inc...

it will vary

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i mean it could just go HTHTHTHT forever and then it would be very close

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or it could go HHHHHHHH forever and then it would be very far

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so it'll be a probability distribution

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so let S_n be the number of heads after n flips

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then (S_n)/n will be the proportion of heads

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and 0.5 will be the expected proportion of heads

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so you're trying to find the distribution of D_n = (S_n)/n - 0.5, the difference in proportion after n tosses

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it turns out that this converges to a normal distribution, by the central limit theorem

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the difference will be normally distributed with mean 0 (equal chance of the proportion being more than c above 0.5 as more than c below 0.5)

and the variance turns out to be s^2/n, where s^2 is the variance of a single toss

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the variance of a single coin flip turns out to 0.25 i believe

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so the variance is 1/4n

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so you can see that as n grows large, the variance gets smaller and smaller

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it's like a thinner and thinner normal distribution

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google like 'central limit theorem coin flipping' for more info, that'll probs help

#probability question