Difference Between Bernoulli and Binomial

Bernoulli vs Binomial

Very often in real life, we come across events, which have only two outcomes that matters. For example, either we pass a job interview that we faced or fail that interview, either our flight depart on time or it is delayed. In all these situations, we can apply the probability concept ‘Bernoulli trials’.

Bernoulli

A random experiment with only two possible outcomes with probability p and q; where p+q=1, is called Bernoulli trials in honor of James Bernoulli (1654-1705). Most commonly the two outcomes of the experiment is said to be ‘Success’ or ‘Failure’.

For example, if we consider of tossing a coin, there are two possible outcomes, which is said to be ‘head’ or ‘tail’. If we are interested in the head to fall; the probability of success is 1/2, which can be denoted as P (success) =1/2, and the probability of failure is 1/2. Similarly, when we roll two dice, if we are only interested in the sum of two dice to be 8, P (Success) =5/36 and P (failure) = 1- 5/36 =31/36.

A Bernoulli process is an occurrence of a sequence of Bernoulli trials independently; therefore, the probability of success remains same for each trial.  In additional, for each trial probability of failure is 1-P(success).

Since the individual trails are independent, probability of an event in a Bernoulli process can be calculated by taking the product of probabilities of success and failure. For an example, if the probability of success [P(S)] is denoted by p and probability of failure [P (F)] is denoted by q; then P(SSSF)=p³q and P(FFSS) = p²q².

Binomial

Bernoulli trials lead to binomial distribution. At most of the occasions, people get confused with the two terms ‘Bernoulli’ and ‘Binomial’.  Binomial distribution is a sum of independent and evenly distributed Bernoulli trials. Binomial distribution is denoted by the notation b(k;n,p); b(k;n,p) = C(n,k)p^kq^n-k, where C(n,k) is known as the binomial coefficient. The binomial coefficient C(n,k) can be calculated by using the formula  n!/k!(n-k)!.

For example, if an instant lottery with 25% winning tickets is sold among 10 people, the probability of purchasing a winning ticket is b(1;10,0.25) = C(10,1)(0.25)(0.75)⁹ ≈ 9 x 0.25 x 0.075 ≈ 0.169