Binomial Distribution is the probability distribution of number of successes in `n`

number of events (trials). For example a die is thrown randomly 10 times, what's the probability of a five on rolling die in 7 out of 10 times? Here probability of a five `(p)`

is (1/6) on a single throw as die has six sides. `n`

is the the number of times die is thrown which is 10. `X`

is the number of times we want success which is 7.

Solution

`P(X < ) `

`P(X > ) `

The sum of all the probabilities shown below will be 1.

- Probability of success must lie between 0 and 1.
- Number of trials can't be less than or equal to 0. It must be a whole number, can't be in decimals.
- Number of successes can't be greater than the number of trials.

See some real world examples where we can use the binomial distribution.

**Quality control:**A factory produces light bulbs, and it is known that 10% of the bulbs are defective. If a sample of 20 bulbs is selected at random, what is the probability that exactly 2 bulbs are defective? This can be modeled using the binomial distribution, with n=20 and p=0.1.**Election polling:**In an election, a candidate has a 60% chance of winning each vote. If a sample of 1000 voters is polled, what is the probability that the candidate will win at least 550 votes? This can be modeled using the binomial distribution, with n=1000 and p=0.6.**Marketing:**A company runs an online ad campaign, and it is known that the click through rate (CTR) is 5%. If the ad is shown to 1000 people, what is the probability that exactly 50 people will click on the ad? This can be modeled using the binomial distribution, with n=1000 and p=0.05.

Binomial Distribution has the following properties -

- Each trial has only two possible outcomes - success or failure.
- The probability of a success on any trial would be same.
- No trials depend on each other.

There is no specific definition of `success`

. You can define it as a desired outcome. It depends on the problem statement. See the examples below -

- If you flip a coin 10 times, what is the probability of getting more than four tails? Here "success" is getting tail.
- Suppose you play a video game (let's say mario). What is the probability that you win 3 of the 10 times. Here "success" is winning the game.
- Suppose school administation wants to know the number of students who take out their names from the randomly selected statistics class. The participation rate in the statistics class is 40% for any given school session. It means withdrawal rate is 60% i.e (1-0.4). Here "success" is a student who took out his/her name.