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Is there a simple formula for finding a value of a cumulative binomial probability, eg. like the ones put in cumulative binomial probability tables? eg. X~B(50, 0.234) Find the cumulative binomial probability for 32, with one equation.

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1 Answer

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I don't know of any shortcut formula for computation. The PDF for $Binom(n, p)$ is $f_X(k) = P(X = j) = {n \choose j}p^j(1-p)^{n-j},$ for $j = 0, 1, \dots, n.$ Then the CDF is $$F_X(k) = P(X \le k) = \sum_{j = 0}^k{n \choose j}p^j(1-p)^{n-j},$$ for $k = 0, 1, \dots, n.$ Also, the CDF can be suitably extended for arguments on the real line.

Printed tables usually show CDF values for ease of use. For example, if $n = 20$ and we want $Q = P(5 \le X \le 15),$ then it is easer to evaluate $Q = 0.9882$ as a difference of two CDF values $F_X(15) - F_X(5)$ than as a sum of eleven PDF values $f_X(5) + f_X(6) + \cdots + f_X(15).$

Most sofware packages have functions for both PDF and CDF. For example, in R, these are dbinom and pbinom, respectively.

 n = 20; p = 1/2 pbinom(15, n, p) - pbinom(4, n, p) ## 0.988182 diff(pbinom(c(4, 15), n, p)) ## 0.988182 sum(dbinom(5:15, n, p)) ## 0.988182

Printed tables are seen less frequently nowadays because software and calculators are more flexible to use. Here are four-place PDF and CDF tables for $Binom(n=10, p=.52),$ which you are unlikely to find in printed form.

 n=10; p=.52; j = 0:10 pdf = dbinom(j, n, p); cdf = pbinom(j, n, p) round(cbind(j, pdf, cdf), 4) j pdf cdf
## [1,] 0 0.0006 0.0006
## [2,] 1 0.0070 0.0077
## [3,] 2 0.0343 0.0420
## [4,] 3 0.0991 0.1410
## [5,] 4 0.1878 0.3288
## [6,] 5 0.2441 0.5730
## [7,] 6 0.2204 0.7933
## [8,] 7 0.1364 0.9298
## [9,] 8 0.0554 0.9852
##[10,] 9 0.0133 0.9986
##[11,] 10 0.0014 1.0000

The specific probability you mentioned is very nearly $1$:

n = 50; p = .234; pbinom(32, n, p)
## 1
n = 50; p = .234; j = 0:50; pdf = dbinom(j, n, p)
plot(j, pdf, type="h", lwd=2, main="PDF of BINOM(50, .234)")
abline(h=0, col="green2")

Note: Following the Comment by @gammatester, for given $n$ and $p$, the CDF $F_X(k) = P(X \le k)$ can be written, in terms of an incomplete beta function, as an integral (transcribing from Wikipedia):

$$ P(X \le k) = I_{1-p}(n-k,k+1) = (n-k){n \choose k}\int_0^{1-p}t^{n-k-1}(1-t)^k\,dt.$$

However, I have not seen this used in basic probability courses for numerical computation. I have not tried it recently, but I seem to recall that, upon evaluating the integral and simplifying, one is back to my displayed equation near the start of this Answer.

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