Mean of Binomial Distribution

The binomial distribution is a common discrete probability distribution model used to describe situations with repeated trials and two possible outcomes. Many students search for the “mean of binomial distribution” because they want a clear understanding of how to find the expected number of successes in such experiments. The mean offers a simple way to predict the average outcome, and it is an important concept in probability. Therefore, this guide explains the meaning of the mean, presents the formula, and shows how the formula is derived. It also provides easy examples to help you calculate the mean step by step.

Mean of a Binomial Distribution: Definition

The mean of a binomial distribution is the expected number of successes in a set of repeated trials. It tells you the average outcome you should expect when the experiment is repeated many times under the same conditions. In a binomial setting, each trial has only two outcomes and the probability of success stays the same. Because of this structure, the mean gives a clear measure of how often success is likely to occur. In simple terms, it shows the long-run average number of successes you can predict from the distribution.

The formula for the mean of a binomial distribution is μ = n × p.

Where:

• μ is the mean or expected value
• n is the total number of trials
• p is the probability of success in one trial

Note. The mean of binomial distribution is also called its expected value, because it shows the average number of successes you can expect over many repeated trials.

Derivation of the Mean (Expected Value)

To understand where the formula μ = n × p comes from, let us look at the binomial distribution in a simple way.

Let X be a binomial random variable such that X ~ Bin(n, p). This means X counts how many successes occur in n repeated trials.

We can think of X as the sum of n small variables, where each small variable represents one trial. Each trial is a Bernoulli trial, which means it has only two outcomes: success or failure. If we call these trials X₁, X₂, …, Xₙ, then:

X = X₁ + X₂ + … + Xₙ

The expected value of one Bernoulli trial is simple:

E(Xᵢ) = p

Now we use the rule called linearity of expectation. It tells us that the expectation of a sum is the sum of the expectations:

E(X) = E(X₁ + X₂ + … + Xₙ) = p + p + … + p (n times)

This gives us:

E(X) = n × p

But, E(X) = μ

Hence, μ = n *p

How to Calculate the Mean of Binomial Distribution: Step-by-step

You can calculate the mean of a binomial distribution in a very simple way. Most students prefer the direct method because it is fast and easy to follow.

1) Main Method: Using the Formula

To calculate the mean of any binomial distribution, we apply the formula, μ = n *p. Here are the steps to calculate the mean:

• Step 1: Identify the number of trials (n).
• Step 2: Identify the probability of success (p).
• Step 3: Multiply n by p.

2) Alternative Method: Using the PMF Table

You can also find the mean by using the probability mass function (PMF) table. This method shows how the mean comes from the full distribution. The process is as follows:

1. List all possible values of X, from 0 to n.
2. Write the probability of each value, P(X = x).
3. Multiply each value by its probability.
4. Add all these products using the formula: μ = Σ[x × P(X = x)]

This method takes more time, but it helps you understand how each outcome contributes to the mean.

Finding the Mean of Binomial Distribution: Examples

Example 1: A class has a quiz with 15 multiple-choice questions. A student knows the correct answer 40% of the time. Each question is independent. Calculate the mean of this binomial distribution.

Solution

From Example 1, we know that:

• The number of trials, n = 15
• The probability of success, p = 0.4

Thus, we can calculate the mean as follows:

μ = n × p

= 15 × 0.4

= 6

Therefore, the student is expected to get 6 questions correct on average. Some scores may be higher or lower, but 6 is the long-run average number of correct answers in this situation.

Example 2: A grocery store inspected crates of tomatoes and recorded the number of rotten tomatoes found in each crate. The results were:

• 95% of crates have 0 rotten tomatoes
• 2% of crates have 1 rotten tomato
• 2% of crates have 2 rotten tomatoes
• 1% of crates have 3 rotten tomatoes

Find the mean number of rotten tomatoes per crate.

Solution

Because this distribution does not follow a standard binomial form, we cannot use μ = n × p. Instead, we use the PMF method. The steps are as follows:

• Step 1: Create a Probability Mass Function Table

• Step 2: Multiply each x by its probability.

0 × 0.95 = 0
1 × 0.02 = 0.02
2 × 0.02 = 0.04
3 × 0.01 = 0.03

• Step 3: Add the products.

0 + 0.02 + 0.04 + 0.03 = 0.09

Thus, the mean number of rotten tomatoes per crate is 0.09. Therefore, on average, each crate contains 0.09 rotten tomatoes. This tells the store that rotten tomatoes are rare, but they still appear occasionally.

Want a Faster Way to Find the Mean? You can use our Binomial Probability Calculator to compute the mean in seconds. It also gives you exact and cumulative probabilities for any binomial distribution.

Conclusion

The mean of a binomial distribution is one of the easiest and most useful ideas in probability. It tells you the expected number of successes in a set of repeated trials, and you can find it quickly using the formula μ = n × p. When the situation does not follow a standard binomial pattern, you can still compute the mean using the PMF method. With these two approaches, you can handle both simple and complex problems with confidence. As you study more probability topics, understanding the mean will help you interpret results and make better predictions.

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