# Binomial Distribution Calculator

Often one would like to model a situation where you have one, and only one, probability assigned to a specific event, such as a dice roll. In these situations, there is a probability of success (p) and the probability of failure (1 - p). In a number of trials (n), one might have a certain number of successes (x), with the rest being failures. In our dice example, you might be playing a game where you lose automatically if you roll two ones with two dice. Maybe you desire to know the probability of rolling this amount just once after, say, 10 attempts. Or maybe you want to know the probabilty of rolling that amount fewer than 3 times in 20 attempts. Or maybe you want to know the probability of rolling that amount between 5 and 8 times in 30 attempts. In any case, the variable number of successes has a binomial distribution if

1. The number of observations (n) is fixed.
2. The observations are independent of each other.
3. Each observation has a probability of success and a probability of failure.
4. The probability of success is the same for each trial.

Under these conditions, the probability of observing x successes in n trials is represented by the probability density function (pdf) P(X = x) = (nCx) px(1 - p)n - x.

The mean of such a distribution is np. The variance is np(1 - p). You can interpret the mean of this distribution as the expected number of successes in n trials.

You can use the calculator below to calculate probabilities under your own scenario using the assumptions you input. The probability density function yields probabilities for a specified number of successes, whereas the cumulative density function yields probabilities for each and every number of successes up to a specified number. The graphs generated are shaped in accordance with the assumptions you enter.