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

\( n \) =
\( p \) =
\( x \) =
\( P(X = x,n,p) \) =
\( P(X \le x,n,p) \) = (at most)
\( P(X \ge x,n,p) \) = (at least)

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