The Binomial distribution is a probability distribution that is used to model the probability that a certain number of “successes” occur during a certain number of trials.

In this article we share 5 examples of how the Binomial distribution is used in the real world.

**Example 1: Number of Side Effects from Medications**

Medical professionals use the binomial distribution to model the probability that a certain number of patients will experience side effects as a result of taking new medications.

For example, suppose it is known that 5% of adults who take a certain medication experience negative side effects. We can use a Binomial Distribution Calculator to find the probability that more than a certain number of patients in a random sample of 100 will experience negative side effects.

- P(X > 5 patients experience side effects) =
**0.38400** - P(X > 10 patients experience side effects) =
**0.01147** - P(X > 15 patients experience side effects) =
**0.0004**

And so on.

This gives medical professionals an idea of how likely it is that more than a certain number of patients will experience negative side effects.

**Example 2: Number of Fraudulent Transactions**

Banks use the binomial distribution to model the probability that a certain number of credit card transactions are fraudulent.

For example, suppose it is known that 2% of all credit card transactions in a certain region are fraudulent. If there are 50 transactions per day in a certain region, we can use a Binomial Distribution Calculator to find the probability that more than a certain number of fraudulent transactions occur in a given day:

- P(X > 1 fraudulent transaction) =
**0.26423** - P(X > 2 fraudulent transactions) =
**0.07843** - P(X > 3 fraudulent transactions) =
**0.01776**

And so on.

This gives banks an idea of how likely it is that more than a certain number of fraudulent transactions will occur in a given day.

**Example 3: Number of Spam Emails per Day**

Email companies use the binomial distribution to model the probability that a certain number of spam emails land in an inbox per day.

For example, suppose it is known that 4% of all emails are spam. If an account receives 20 emails in a given day, we can use a Binomial Distribution Calculator to find the probability that a certain number of those emails are spam:

- P(X = 0 spam emails) =
**0.44200** - P(X = 1 spam email) =
**0.36834** - P(X = 2 spam emails) =
**0.14580**

And so on.

**Example 4: Number of River Overflows**

Park systems use the binomial distribution to model the probability that rivers overflow a certain number of times each year due to excessive rain.

For example, suppose it is known that a given river overflows during 5% of all storms. If there are 20 storms in a given year, we can use a Binomial Distribution Calculator to find the probability that the river overflows a certain number of times:

- P(X = 0 overflows) =
**0.35849** - P(X = 1 overflow) =
**0.37735** - P(X = 2 overflows) =
**0.18868**

And so on.

This gives the parks departments an idea of how many times they may need to prepare for overflows throughout the year.

**Example 5: Shopping Returns per Week**

Retail stores use the binomial distribution to model the probability that they receive a certain number of shopping returns each week.

For example, suppose it is known that 10% of all orders get returned at a certain store each week. If there are 50 orders that week, we can use a Binomial Distribution Calculator to find the probability that the store receives more than a certain number of returns that week:

- P(X > 5 returns) =
**0.18492** - P(X > 10 returns) =
**0.00935** - P(X > 15 returns) =
**0.00002**

And so on.

This gives the store an idea of how many customer service reps they need to have in the store that week to handle returns.

**Additional Resources**

6 Real-Life Examples of the Normal Distribution

5 Real-Life Examples of the Poisson Distribution