# Probability of binomial n success before m failures?

problem of n success before m failures where binomial probability of success is p has a standard textbook solution as follows $$P = \sum_{k=n}^{m+n-1} \binom{m+n-1}k p^k (1-p)^{m+n-1-k}$$

I am however unable to come up with this solution and i am not sure where i deviate and how?

(a) last trial has to be a success => probability of p

(b) in t-1 trials before last trial, n-1 have to be success and (t-1)-(n-1) failures => probability of $\binom{t-1}{n-1} p^{n-1} (1-p)^{t-n}$

Combining (a) and (b) gives me probability that t trials have t-n failures before n success $P_{t} = \binom{t-1}{n-1} p^{n} (1-p)^{t-n}$ I understand that this is negative binomial pmf as well.

From here on, i say that total probability is sum of all the probabilities with various t i.e. $P = \sum_{n}^{m+n-1} P_{t}$ that can be written as $$P = \sum_{t=n}^{m+n-1} \binom{t-1}{n-1} p^{n-1} (1-p)^{t-n}$$

I believe something is off with my last step.. may be these events of different $t$ trials are not mutually exclusive but i am not able to see it.

For example: n = 3 success and m = 2 failures; p = binomial probability

(a) I can have a 3 trial solution SSS with probability $p^3$

(b) 4 trial solution {SSFS,SFSS,FSSS} with probability $\binom{3}{2} p^{3} (1-p)^1$

(c) 5 trial solution is not possible since n=2 would have happened

I would therefore add probabilities from (a) and (b) as solution but that would give $$P = p^3 + \binom{3}{2} p^{3} (1-p)^1$$

and of course, this is not right. Can someone help and point out why this is not correct and what can i fix here?

• This appears to be a question about a negative binomial random variable. There several versions of negative binomial distributions. Please describe the random variable and carefully define $m, n, p.$ Apr 2, 2018 at 2:00
• I have added some of these details. Please see if you can answer it now? Apr 2, 2018 at 2:11
• I found another link to similar problem. Listing here. math.stackexchange.com/questions/915353/… Jun 13, 2020 at 19:04

The book answer (or your transcription of it) appears to be incorrect

By the way, it is simpler to count failures, and look at the results in reverse.
With $0 \le k < m$, the last trial must be a success, and the $k$ failures can be distributed any which way in the remaining $(k+n-1)$ trials, thus

$$P = \sum_{k=0}^{m} \binom{k+n-1}k p^n (1-p)^k$$

If you want to count successes (as the book has done), you should now be able to correct the formula you have transcribed.

I actually came across this similar problem in Bertsekas' "Introduction to Probability" 2nd edition (Ch 6 Exercise 4c), where it was making the same mistake. May I ask which textbook are you referring?

## Original formula:

$$P = \sum_{k=n}^{m+n-1} \binom{m+n-1}k p^k (1-p)^{m+n-1-k}$$ The problem of this original formula is that

1) as you vary the iterating variable k, the total number of trials should also be changing, but the term $$m+n-1$$ is kept constant

2) the failure exponent term $$m+n-1-k$$ will be counting the right number of failure.

With (1) and (2) above, this original formula is actually accumulating the probability of getting k success and the rest failure while running m+n-1 trials for the range of $$k>=n$$. This is totally not what the question is asking.

I would say the following: minor correction to @{true blue anil}'s formula: $$P = \sum_{k=0}^{m-1} [\binom{k+n-1}k p^{n-1} (1-p)^{k} ]p = \sum_{k=0}^{m-1} \binom{k+n-1}k p^n (1-p)^{k}$$ This is saying that we want to accumulate within the range $$0<=k<=m-1$$, the probability that there are n successes (last trial is forced to be success, the first n-1 success happen within in the k+n-1 trials, the rest of the k trials are failure).

## For you example,

n = 3 success, m = 2 failures, p = prob of success

### If we do it by hand,

3 trials: XXP => $$p^3$$

4 trials: XXXP => $$\binom{3}{2}p^2 (1-p) p = \binom{3}{2}p^3 (1-p)$$

$$P = p^3 + \binom{3}{2}p^3 (1-p) = p^3 +3p^3(1-p)$$

### And if you apply the formula I have above: $$\sum_{k=0}^{m-1} \binom{k+n-1}k p^n (1-p)^{k}$$

You would be iterating k = 0 to 1.

So $$P = \binom{0+3-1}0 p^3 (1-p)^{0} + \binom{1+3-1}1 p^3 (1-p)^{1} = p^3 + 3p^3(1-p)$$, which should agree with the solution by hand.

• Is this your belief that correct formula should be as noted below by you or have your cross referenced other books and literature. You seem to be correct but i wanted to check before i mark your answer as final answer to question. Jun 12, 2020 at 23:41
• I feel fairly comfortable that the steps that I am providing should be right. I am asking you where you came across this problem, so that I can trace if they came from the same source. Jun 13, 2020 at 2:33
• unfortunately, this question was posed by me more than 2 years ago. I wish i had made a note / refer to book in original post. I dont have recollection of it any more. Jun 13, 2020 at 18:54
• i also found another link to similar problem on stack exchange. Should merge it.math.stackexchange.com/questions/915353/… Jun 13, 2020 at 19:03