Binomial distribution: counting successes in random trials

In short: The binomial distribution describes how likely it is to get exactly k successes in n identical, independent trials. Typical examples are coin tosses, series of dice rolls or quality checks.

In more detail: If each trial has only the outcomes success (probability p) or failure (1 − p), then P(X = k) = (n choose k) · p^k · (1 − p)^(n − k). The binomial coefficient (n choose k) = n! / (k! · (n − k)!) counts the number of ways the k successes can be spread over the n trials.

In detail

• Conditions: The number of trials n is fixed, each trial has exactly two outcomes, the probability of success p stays the same and the trials do not influence each other. A single trial of this kind is called a Bernoulli trial.
• Coin example: With three tosses of a fair coin, the probability of exactly two heads is (3 choose 2) · 0.5² · 0.5 = 3 · 0.125 = 0.375, i.e. 37.5 %.
• Dice example: The probability of rolling at least one six in four rolls is calculated using the complementary event: 1 − (5/6)^4 ≈ 1 − 0.482 = 0.518, i.e. just over 50 %.
• Expected value: On average you expect E(X) = n · p successes. With 60 rolls you therefore expect about 60 · 1/6 = 10 sixes.
• Spread: The variance is n · p · (1 − p), and the standard deviation is its square root. With 60 rolls that is about the square root of 8.33, roughly 2.9 sixes.
• Shape of the distribution: With p = 0.5 the distribution is symmetrical. With a small p it is skewed to the right, because few successes are the most common outcome.
• Without replacement: If you draw from a small set without replacement, p changes from draw to draw. The hypergeometric distribution is then a better fit.
• Approximation: For large n the binomial distribution looks more and more like a bell curve. As a rule of thumb, approximating it by the normal distribution is considered acceptable when n · p · (1 − p) is greater than 9.
• Context: The binomial distribution links combinatorics with probability theory and leads on to the normal distribution (kombinatorik-grundlagen, wahrscheinlichkeitsrechnung-grundlagen, normalverteilung).