Binomial Distribution

The binomial distribution is a model which allows a representation of the average number of successes obtained with a repetition of successive independent trials.

$$ P(X=k) = {n \choose k} \, p^{k} (1-p)^{n-k} $$

with $ k $ the number of successes, $ n $ the total number of trials/attempts/expériences, and $ p $ the probability of success (and therefore $ 1-p $ the probability of failure).

The binomial distribution can be used in situations with 2 contingencies (success or failure, true or false, toss or tails, etc.) that can be repeated and independent.

Example: Calculation of the probability to draw 4 times the number 6 after 5 successive dice rolls: the probability $ p $ to make a 6 is $ 1/6 $, the total number of trials is $ n = 5 $, the total number of successes expected is $ k = 4 $. $$ P(X=4) = {5 \choose 4} \, \left(\frac{1}{6}\right)^4 \left(1-\frac{1}{6}\right)^{5-4} = {5 \choose 4} \left(\frac{1}{6}\right)^4 \left(\frac{5}{6}\right)^1 = \frac{5^2}{6^5} \approx 0.00321 \approx 0.3% $$

The formula for the binomial distribution involves the binomial coefficient $ {n \choose k} $ (which can be read as a combination of $ k $ among $ n $).