"Help please= Calculating H hits from D dice" Topic

I'm a lot better than your average Joe with probabilities, but I can't work out a formula for this.

I am Rolling D number of dice. The chance of one success (hit) is C. I heed to get H number of hits to win.

eg Hit is 5 or 6 =1/3. I need to get 2 hits from 4 dice.

What's the formula to calculate the over all chance. I can do it if the hits required are 1 or d-1, but not for numbers inbetween (Though 50/50 I can imagine a binary table in my head if the number of misses permitted is low enough, but I want a maths formula)

Working out the inverse is often easier.

i.e. in your example 2, 3, or 4 hits from 4 would be OK. So the only failures are1 or less hits. 4 misses is 2/3 ^ 4, and 1 hit is 1/3 x 2/3 ^3.

0.197 + 0.098 = 0.2957 chance of failure
So chance of success is 0.7043

Fred is close.

He missed the fact that the 1 hit can be any of the 4 dice not just the first.

so the chance of failure, F, where D – H = 2 is

F = (1-C)^D + (D choose 1) * C * (1-C)^(D-1)

(D choose 1) is a combination. (see link ) (n choose r) is usually written C(n,r) but since you are using C, I didn't want to confuse things.
Anyway, (n choose r) = n! / (r! * (n-r)!). As you can see:(n choose 1) = n.

so this can be simplified to

F = (1-C)^D + D * C * (1-C)^(D-1)

Binomial distribution gives the probability of the required number of successes in n trials (= number of dice), given the probability of a single success.

there's an easy way to do the calculation – if you have MS Excel, set up a table using the function BINOMDIST. Just say and I can e-mail you a copy of a working spreadsheet.

Of course you could calculate the chance of success, S directly:

S = (D choose H) * C^H * (1-C)^(D-H) +
(D choose H+1) * C^(H+1) * (1-C)^(D-(H+1)) +
… +
(D choose D) * C^D * (1-C)^(D-D)

[That last term:"(D choose D) * C^D * (1-C)^(D-D)" just equals "C^D"]

Here is an online binomial calculator: link

Just put in n = D, p = C, and Prob X is at least H

Probably a bad example, as only one failure needed- What If I am looking for 4 out of 7.

Tanuki- I am using Open Office, but that seems to have virtually all the MS functions (plus a few extra).

IanMHopping (at) AOL (dot) com

Thanks for your help

Last Hussar – will drop you the spreadsheet this evening (I'm at work just now).

Would You email the spreadsheet my way too!

OW – drop me an email at tanuki_derek at yahoo dot co dot uk and I'll mail it off to you!

Spreadsheet sent chaps – let me know if you don't receive it!