Quick Binomial Distribution Calculator

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.

Probability of Success (p) Number of Trials (n) Number of Successes (X)

Solution

$P(X=7) = 0.000242$ $P(X \lt7) = 0.999739$ $P(X \le7) = 0.999981$ $P(X \gt7) = 0.000019$ $P(X \ge7) = 0.000261$

P(X < 7) means probability of less than 7 successes. Similarly P(X > 7) refers to probability of more than 7 successes.

How to calculate P(X < 7)

To calculate P(X < 7), we need to sum all the probabilities from P(X = 0) through P(X = 7). See the details below.

$P(X \lt7)$ = P(X=0) + P(X=1) + P(X=2) + P(X=3) + P(X=4) + P(X=5) + P(X=6)

$P(X \lt7)$ = 0.162802 + 0.324043 + 0.290240 + 0.154052 + 0.053660 + 0.012817 + 0.002126

$P(X \lt7) = 0.999739$

How to calculate P(X > 7)

To calculate P(X > 7), we need to sum all the probabilities from P(X = 8) through P(X = 10). See the details below.

$P(X \gt7)$ = P(X=8) + P(X=9) + P(X=10)

$P(X \gt7)$ = 0.000018 + 0.000001 + 0.000000

$P(X \gt7) = 0.000019$

Binomial Distribution Table

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

$P(X=0) = 0.162802$ $P(X=1) = 0.324043$ $P(X=2) = 0.290240$ $P(X=3) = 0.154052$ $P(X=4) = 0.053660$ $P(X=5) = 0.012817$ $P(X=6) = 0.002126$ $P(X=7) = 0.000242$ $P(X=8) = 0.000018$ $P(X=9) = 0.000001$ $P(X=10) = 0.000000$

Important Points

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.

Examples of Binomial Distribution

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.

FAQs

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.