Help with binomial probability for roulette?

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August 20th, 2019 at 3:49:27 AM permalink

Maybe use the Sterling approximate for a factorial? Maybe it would allow simplification of the fraction before actually computing 171!

The problem is you ask the software to compute a huge number (171!) to get a much smaller one.

Note: Excel computes COMBIN(171,x) without problem.

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Ace2

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August 20th, 2019 at 9:35:49 AM permalink

The factorials cancel.

No need to use excel for something that can be done by any calculator or even by hand or slide rule.

(36/37)^169 * 170/37 = 0.0448 = a

a * 36/37 * 171/170 = 0.0438

In 171 spins you expect to hit your 1/37 number 4.62 times. You have a 1 - (36/37)^171 = 99.08 % chance of hitting it at least once, a 4.38% chance of hitting it exactly once and a 94.7 % chance of hitting it twice or more

Last edited by: Ace2 on Aug 20, 2019

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CrystalMath

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August 20th, 2019 at 9:51:40 AM permalink

Try this:


	double binomDist(int x, int n, double p)
	{
		double q = 1.0-p;
		
		double result = 1.0;
		for(int d = n+1-x; d<=n; d++)
		{
		result*=d;
		}
		result*=Math.pow(p,x);
		result*=Math.pow(q,n-x);
		return result;
	}

It avoids getting huge numbers because most terms cancel.

I heart Crystal Math.

MichaelBluejay

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August 20th, 2019 at 12:08:32 PM permalink

Thanks everyone for the help!

@OnceDear, yes, I should have figured out that one of my variables besides the actual result was going out of bounds. (Duh.)

@CrystalMath, your solution is perfect and exceptionally helpful. Thanks so much!

I run Easy Vegas ( https://easy.vegas )

MichaelBluejay

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August 21st, 2019 at 6:18:52 AM permalink

Okay, I'm stuck again.

So, my understanding of canceling the factorials doesn't jibe with CrystalMath's Java/C code above. (CrystalMath, I'm referring to you in the third person not out of disrespect, but because other people might be helping besides you.)

So I must be missing something. Here's my understanding:

Binomial coefficient:

    n!
-----------
(n-x)! * x!

n = 10
x = 7

n! = 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2
x! =              7 x 6 x 5 x 4 x 3 x 2
(n-x)! =                          3 x 2

So, to do the most cancelation, I should cancel n! with x!, not with (n-x)!

The binomial coefficient formula can be rewritten as:

    n!         n!       1      
----------  = ---- *  ------
(n-x)! * x!    x!     (n-x)!

So, I'll calculate the factorials after canceling, then divide by the factorial of n-x, then multiply by the rest of the binomial probability formula.

After cancelation, I'll be multiplying 8 x 9 x 10. So, my code starts with:

for (i=x-0+1; i <= n; x++) { // Starts at 8, finishes at 10

CrystalMath's code is:

for(int d = n+1-x; d<=n; d++) // Starts at 4, ends at 10

So, that's the first thing that doesn't make sense to me. I'm guessing he's taking a shortcut.

However, if I use his code, looking for the probability of at least one hit of a single number over 37 spins, I get 38%. That does jibe with what I get in my spreadsheet. But it's not intuitive (seems low, seems like it should be >50%), and it doesn't jibe with the formula:

1 - (ways to win / total ways) ^ trials = 1 - (36/37)^37 = 64%.

So, that's the second thing I'm confused about.

Finally, when I use my own JS code, my results are even farther off. My understanding for "at least" is that I find the probability for exactly x matches, then multiply it by the number of trials.



p = 1/37;
x = 1;
n = 37;

sum = 0;
for (hitcounter=1; hitCounter <= x; hitCounter++) {
	oneRun = probExactlyXhitsInNtrials(p, hitCounter, n);
	sum += oneRun * n;
}
showResult(sum);

function probExactlyXhitsInNtrials(p, x, n) {
	result = 1;
	for (i=x-0+1; i<=n; i++) result *=i; // This line differs from CM's version
	result /= fact(n-x);                 // This line doesn't appear in CM's version
	result *= Math.pow(p,x);
	result *= Math.pow(q,n-x);
	return result;
}

That gives me probability of <0.1%, which is clearly wrong. That's the third thing I'm lost on.

I run Easy Vegas ( https://easy.vegas )

kubikulann

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August 21st, 2019 at 6:35:42 AM permalink

1st thing.
There is a symmetry : C_n^x = C_n^n-x
So you could put some code like x:= max(x,n-x)?

2d thing and 3d thing.
P(at least) is not P(exactly), but it is also not [1-P(exact)]^n or *n. You need to compute P(exact) for all the x�s from x to n, then sum it.

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CrystalMath

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August 21st, 2019 at 6:58:14 AM permalink

I see, I wasn't dividing by the x! term.

Looking at this example:

Quote: MichaelBluejay


n! = 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2
x! =              7 x 6 x 5 x 4 x 3 x 2
(n-x)! =                          3 x 2

We can simplify this to 10 x 9 x 8/(3 x 2 x 1). So, the loop should just do three steps.

I've modified my code below.


	double binomDist(int x, int n, double p)
	{
		double q = 1.0-p;

		double numerator = n;
		double result = 1.0;
		for( int i=x; i>=1; i-- ) result *= numerator--/i;

		result*=Math.pow(p,x);
		result*=Math.pow(q,n-x);
		return result;
	}

If you want "at least," you will need to loop through all possibilities and sum them.

This gives the cumulative distribution from 0 hits up to x. If you want x or greater, then calculate 1 - cumulativeBinomDist(x-1,n,p).


	double cumulativeBinomDist(int x, int n, double p)
	{
		double result = 0.0;
		for(int i = 0; i <= x; i++)
		{
			result += binomDist(i, n, p);
		}
		return result;
	}

I heart Crystal Math.

kubikulann

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August 21st, 2019 at 7:06:57 AM permalink

...?
The problem with blocking members is that you end up posting redundant info!

I sometimes feel nobody reads my posts.

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7craps

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August 21st, 2019 at 8:41:26 AM permalink

for a working example of a calculator that uses javascript online for binomial probabilities, I like this one.
https://www.di-mgt.com.au/binomial-calculator.html

It gives lots of info at one time (shows a table) and has those funny to see negative values that who knows how accurate it is.
It only accepts a decimal value for p (frustrating for many)
where many other javascript sites accept a decimal or fraction.

this one has been around for a long time and prefers a fraction over a decimal for p.
http://vassarstats.net/binomialX.html

winsome johnny (not Win some johnny)

MichaelBluejay

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August 21st, 2019 at 3:07:59 PM permalink

Quote: kubikulann
...?
The problem with blocking members is that you end up posting redundant info!

I sometimes feel nobody reads my posts.

kubikulann, I haven't blocked you, I just either don't understand what you're saying, or it doesn't seem to apply to my questions. I do appreciate your trying to help.

I run Easy Vegas ( https://easy.vegas )

gordonm888
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August 21st, 2019 at 7:05:08 PM permalink

Quote: kubikulann
...?
The problem with blocking members is that you end up posting redundant info!

I sometimes feel nobody reads my posts.

I read all your posts. You are an excellent addition to our forum!

So many better men, a few of them friends, are dead. And a thousand thousand slimy things live on, and so do I.

kubikulann

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August 21st, 2019 at 7:51:20 PM permalink

Quote: MichaelBluejay
kubikulann, I haven't blocked you, I just either don't understand what you're saying, or it doesn't seem to apply to my questions. I do appreciate your trying to help.

I said that the combination of x among n is equal to the combination of (n-x) among n.

n! / x! n-x! = n!/ n-x! x!

So to reduce size you can choose to simplify n! with the largest of x! or n-x!

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7craps

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August 21st, 2019 at 8:42:46 PM permalink

Quote: kubikulann
I said that the combination of x among n is equal to the combination of (n-x) among n.

n=37;
k=35;
binomial(n,k) == binomial(n,n-k)
binomial(37,35) == binomial(37,2) ?

https://www.wolframalpha.com/input/?i=binomial%2837%2C35%29

https://www.wolframalpha.com/input/?i=binomial%2837%2C2%29

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kubikulann

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August 22nd, 2019 at 6:23:25 AM permalink

Quote: 7craps
n=37;
k=35;

binomial(37,35) == binomial(37,2) ?

No !
combin(37,35)==combin(37,2)
but binomial is combin*p³⁵q²< > combin*p²q³⁵

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7craps

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August 22nd, 2019 at 6:38:12 AM permalink

Quote: kubikulann
No !
combin(37,35)==combin(37,2)
but binomial is combin*p³⁵q²< > combin*p²q³⁵

combin(37,35)==binomial(37,2)
Looks like you are NOT the expert you think you are.
that is so sad, HUMAN!

binomial(n,k) is a function used for combinations in many languages
Wolfram Alpha and pari/gp use it, just a couple that come to mind.

gp > binomial(37,2)
%1 = 666

that you for sharing you FALSE opinion on the subject

winsome johnny (not Win some johnny)

kubikulann

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August 22nd, 2019 at 6:46:10 AM permalink

7craps, go to bed and drain that alcohol from your brain, then come back to talk sense. (If you can, that is...)

What should you make of a subhuman calling you human? I know, they suffer from their impropriety, they need to lash out at people who are not as limited as they are. They can�t stand their inferiority, so they blame the normal people for being relatively superior to them.
On the other hand, if they call themselves �crap�...

Oignez vilain, il vous poindra.
Poignez vilain, il vous oindra.

Don�t give pearls to pigs; they�ll bite you.

Some people are plain stupid.
Others are just mean.
But being mean AND stupid labels you for the dustbins of society.

Last edited by: kubikulann on Aug 22, 2019

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7craps

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August 22nd, 2019 at 7:17:43 AM permalink

Quote: kubikulann
Yes I do HATE such kind of FUCKING MORON as 7craps is.

OH.
imo, YOU ARE THE EXPERT WAY MORE THAN WOLFRAM ALPHA IS.
GREAT! that is SUPER DUPER

Wolfram Language code:
Binomial[37, 2]
Result: 666

LOL

you just got more to learn, in my opinion
I like to learn, so thanks for sharing your FALSE information
nice to know what is NOT correct.

winsome johnny (not Win some johnny)

kubikulann

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August 22nd, 2019 at 7:22:19 AM permalink

Try binomial[37,35], mo� 7craps.

(Note: if I call someone else « 7craps », is it considered a personal insult?)

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OnceDear

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August 22nd, 2019 at 7:43:50 AM permalink

Quote: kubikulann
Try binomial[37,35], mo� 7craps.

(Note: if I call someone else « 7craps », is it considered a personal insult?)

No. But your previous post was undoubtedly an insult.
See ya later.

Psalm 25:16 Turn to me and be gracious to me, for I am lonely and afflicted. Proverbs 18:2 A fool finds no satisfaction in trying to understand, for he would rather express his own opinion.

MichaelBluejay

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August 22nd, 2019 at 4:31:57 PM permalink

Quote: kubikulann
What should you make of a subhuman calling you human? I know, they suffer from their impropriety, they need to lash out at people who are not as limited as they are. They can�t stand their inferiority...But being mean AND stupid labels you for the dustbins of society.

kubikulann blocked.

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Boz

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August 22nd, 2019 at 5:38:00 PM permalink

Quote: MichaelBluejay
kubikulann blocked.

Why do you and others have to publicly announce you �Blocked� someone? Are you proud? Does ANYONE care? Just asking.

Odds are you have me blocked so you won�t see this.

But it appears to me you came back to AGAIN to push a money making venture on your end. Like Norm, you come back again and again when you have an agenda. Face it, you are a capitalist, like most of us.

You may hide under the Tiny House living, bike riding �environmentalist� label. But at the end of the day you still use this site to push your business ventures.

While I legally burn $13.99 cases of foam takeout containers in your name every �Earth Day�, you still push your agenda. I just wish I knew what it was.

Either way, I�ll keep flying extra segments across the country. Because it�s saves me money, gets me more miles and better status. If it upsets hippies, that�s a bonus.

And if anything I posted �triggers� you, I�m sure you will report it to The Wizard and demand a suspension or apology like you did when I claimed you were a closeted Trump supporter. Obviously you thought it may hurt the �business� you have going.

PS, sorry about your upbringing, no one should ever have to go through what you did Mr BJ.

MichaelBluejay