The binomial distribution is a probability distribution that is used to model the probability that a certain number of “successes” occur during a fixed number of trials.
The binomial distribution is appropriate to use if the following three assumptions are met:
Assumption 1: Each trial only has two possible outcomes.
We assume that each trial only has possible two outcomes. For example, if we flip a coin 100 times, each time there can only be two possible outcomes – heads or tails.
Assumption 2: The probability of success is the same for each trial.
We assume that the probability of achieving a “success” is the same for each trial. For example, the probability of a coin landing on heads is 0.5 for any given flip. This probability does not change from one coin flip to the next.
Assumption 3: Each trial is independent.
We assume that each trial is independent of every other trial. For example, the outcome of one coin flip does not affect the outcome of another coin flip. The flips are independent.
The following examples show various scenarios that meet the assumptions of the binomial distribution.
Example 1: Number of Free Throws Made
Suppose a basketball player is known to make 70% of his free throws attempts. If he makes 20 attempts, this scenario can be modeled using the binomial distribution.
This scenario meets each of the assumptions of the binomial distribution:
Assumption 1: Each trial only has two possible outcomes.
For each free throw attempt, there are only two possible outcomes – a make or a miss.
Assumption 2: The probability of success is the same for each trial.
The probability that the player makes a free throw on each attempt is the same – 70%. This does not change from one attempt to the next.
Assumption 3: Each trial is independent.
Each free throw attempt is independent of every other attempt. Whether or not a player makes one attempt does not affect whether he makes another attempt.
Example 2: Number of Side Effects
Suppose it’s known that 5% of adults that take a certain medication experience negative side effects. Suppose a medical profession then gives this medication to 100 adults in a given month.
This scenario meets each of the assumptions of the binomial distribution:
Assumption 1: Each trial only has two possible outcomes.
For each adult that receives the medication, there is only two possible outcomes – they experience negative side effects or they do not.
Assumption 2: The probability of success is the same for each trial.
The probability that each adult experiences a negative side effect is the same – 5%.
Assumption 3: Each trial is independent.
The outcome for each adult is independent. Whether or not one adult experiences negative side effects does not affect whether or not another adult does as well.
Example 3: Number of Shopping Returns
Suppose it’s known that 10% of all customers who walk into a shop are there to make a return. Suppose 200 people enter a store in a given day and the manager records the number who are there to make a return.
This scenario meets each of the assumptions of the binomial distribution:
Assumption 1: Each trial only has two possible outcomes.
Each time a customer enters the shop, there are only two reasons they may be there – to make a return or not.
Assumption 2: The probability of success is the same for each trial.
The probability that a given customer is there to make a return is the same – 10%.
Assumption 3: Each trial is independent.
The outcome for each customer is independent. Whether or not one customer is there to make a return does not affect whether or not another customer is there to make a return.
Additional Resources
The following tutorials offer additional information on the binomial distribution:
An Introduction to the Binomial Distribution
Binomial Distribution Calculator
5 Real-Life Examples of the Binomial Distribution