The 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 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 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 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 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 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.