# Binomial Probability Distribution Calculator

An online calculator to calculate the binomial probability distribution and the probabilities of "at least" and "at most" related to the binomials.

## Binomial Probability Distribution

If in a given binomial experiment, the probability that in a single trial event A occurs is $p$, then the probability that A occurs exactly $x$ times in $n$ trials is given by:
$P(X = x,n,p) = {n \choose x} \cdot p^x \cdot (1-p)^{n-x} = \dfrac{n!}{x! (n-x)!} \cdot p^x \cdot (1-p)^{n-x}$
The calculator below calculates the binomial probability distribution $P(X = x,n,p)$ from $x=0$ to $x = n$, for different values of n and the probability p. The calculator below helps in investigating these distributions in various situations.
The same calculator also calculates the probability of "at least" $x$ given by $P(X \ge x,n,p)$ and "at most" $x$ given by $P(X \le x,n,p)$

Example 1
At each trial, the probability that event A occurs is $p = 0.4$
a) What is the probability that event A occurs 3 times after 6 trials?
b) What is the probability that event A occurs at least 3 times after 6 trials?
c) What is the probability that event A occurs at most 3 times after 6 trials?

Solution to Example 1
a) $P(X = 3,6,0.4) = \dfrac{6!}{3! (6-3)!} \cdot 0.4^3 \cdot (1-0.4)^{6-3} = 0.276480$
b) At least 3 times means $x$ is either $3, 4, 5 \; \text{or} \; 6$ or $x \ge 3$
$P(\text{at least 3 times}) = P( X = 3 \; or \; X = 4 \; or \; X = 5 \; or \; X = 6 )$
Using the binomial formula, the probability may be written as
$P(X \ge 3,6,0.4) = P(X = 3,6,0.4) + P(X = 4,6,0.4) + P(X = 5,6,0.4) + P(X = 6,6,0.4) = 0.455680$
c)
At most 3 times means $x$ is either $0, 1, 2 \; \text{or} \; 3$ or $x \le 3$
$P(\text{at most 3 times}) = P( x = 0 \; or \; x = 1 \; or \; x = 2 \; or \; x = 3 )$
Using the binomial formul, the probability may be written as
$P(X \le 3,6,0.4) = P(X = 0,6,0.4) + P(X = 1,6,0.4) + P(X = 1,6,0.4) + P(X = 3,6,0.4) = 0.820800$

## How to use the calculator

1 - Enter $n$ and $p$ and $x$ and press "calculate". $n$ and $x$ are positive integers and $p$ real satisfying the conditions:
$0 \lt p \lt 1$ , $n \ge 1$ , $0 \le x \le n$

 $n$ = 4 $p$ = 0.1 $x$ = 3 $P(X = x,n,p)$ = $P(X \le x,n,p)$ = (at most) $P(X \ge x,n,p)$ = (at least)

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