# 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:

\( 0 \lt p \lt 1 \) , \( n \ge 1 \) , \( 0 \le x \le n \)

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