Binomial distribution

< List of probability distributions

There’s a lot of situations in life where we can be faced with two possible outcomes. For example, you either win the lottery or you don’t and a drug to cure a disease works or it doesn’t. Binomial distributions come into play. When analyzing the probability of success or failure such as these. This type of distribution evaluates the likelihood for an experiment or survey outcome to either succeed or fail – such as coins landing on heads-or-tails during tosses or passing/failing tests.

Binomial distribution formula

Your experiment must meet the following three criteria in order to use the formula below:

  1. Fixed number of observations or trials. In other words, you can have 10 trials or 100, but not an unlimited number of attempts.
  2. Independent observations or trials. In other words, one coin toss or test attempt (or whatever it is you are measuring) should not affect the next attempt.
  3. Same probability of success from one “trial” to another. for example, the odds of flipping heads remains at 50% from trial to trial.
How to figure out if an experiment is binomial.

The binomial distribution formula is:

b(x; n, P) = nCx * Px * (1 – P)n – x


  • b = binomial probability.
    x = total number of “successes.”
    P = probability of a success on a single attempt.
    n = number of attempts or trials.
Bernoulli distribution formula example.

The binomial distribution formula can also be written as:

binomial distribution formula
Alternate formula with factorials.

Example (using alternate formula): A fair coin is tossed 10 times. What is the probability of getting six heads?

  • Number of trials (n) = 10
  • Odds of success (“heads”) is 50% or 0.5 (1 – p = 0.5)
  • x = 6 (we want to know the probability of getting 6 heads)
  • P(x=6) = 10C6 * 0.5^6 * 0.5^4 = 210 * 0.015625 * 0.0625 = 0.205078125 = 20.51%.

Relationship to Bernoulli Distribution

The binomial distribution is closely related to the Bernoulli distribution: the binomial distribution with n = 1 is called a Bernoulli distribution. In addition, if a Bernoulli trial has independent trials, the number of successes follows a binomial distribution.  

A Bernoulli distribution is a set of Bernoulli trials. Each of these trials has one possible outcome, Success, or Failure. In each trial, the probability of success, P(S) = p, is the same. The probability of failure is 1 minus the probability of success: P(F) = 1 – p. (Remember that “1” is the total probability of an event occurring and is always between zero and 1). Finally, all Bernoulli trials are independent from each other and the probability of success doesn’t change from trial to trial, even if you have information about the other trials’ outcomes.