Binomial Distribution

The binomial distribution is a model (a law of probability) which allows a representation of the average number of successes (or failures) 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.

Exercises involving the binomial distribution look like:

— A person randomly draws a card 10 times from a deck of 52 cards. What is the probability that she draws at least 7 red color cards?

— A basketball player hits 60% of his free throws. If the player attempts 10 free throws, what is the probability that he will hit exactly 8?

— A merchant has a conversion rate of 10% on his online sales site. If 200 people visit his site, what is the probability that 50% will make a purchase?

— A medicine has a 75% success rate in relieving an illness. If given to 10 patients, what is the probability that at least 8 patients will feel better after taking the drug?

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

The expectation (the average of the positive results) is equal to $ E(X) = n \times p $

The variance is equal to $ V(X) = n p (1-p) $