Binomial probability shows the chance of getting a certain number of successes in a fixed number of trials. Each trial has only two outcomes: success or failure. For example, it can help you find the chance of getting three heads in five coin tosses. It can also show the chance of finding two defective bulbs in a batch of ten. Binomial probability is part of the wider Binomial Distribution, which explains how these outcomes are spread. This concept is widely used in statistics, research, and everyday problem-solving. In this guide, you will learn how to calculate binomial probability step by step using clear and simple examples.
What is a Binomial Experiment?
A binomial experiment is a statistical experiment that has a fixed number of independent trials. Each trial in a binomial experiment has only two possible outcomes: “success” or “failure.” Additionally, the probability of success is the same for every trial. This type of experiment is common in statistics and everyday situations where we want to calculate the chance of a specific number of successes.
In other words, an experiment qualifies as binomial if it meets these conditions:
- Fixed number of trials: The experiment is repeated a set number of times.
- Two possible outcomes: Each trial results in either success or failure.
- Constant probability of success: The chance of success stays the same for every trial.
- Independent trials: The outcome of one trial does not affect the others.
Examples of binomial experiments include:
- Flipping a coin several times and counting the heads.
- Testing a batch of bulbs to see how many are defective.
- Checking how many customers make a purchase from a fixed number of visitors.
The Binomial Probability Formula
The binomial probability formula helps us calculate the chance of getting a specific number of successes in a fixed number of trials. The formula is:
$P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$
Where:
- n is the total number of trials in the experiment.
- k is the number of successes we wish to estimate
- p is the probability of success on a single trial.
- 1 − p is the probability of failure on a single trial.
- $\binom{n}{k}$ is the number of different ways to get exactly k successes in n trials.
Step-by-Step Guide: How to Calculate Binomial Probability
Calculating binomial probability may seem tricky at first. However, if you follow a few simple steps, it becomes very easy. You can even extend the same steps to calculate cumulative probabilities, which give the chance of getting up to a certain number of successes.
Here is the step-by-step process of finding any binomial probability:
- Identify the values corresponding to the total number of trials, n, number of successes you wish to calculate, k and the probability of success, p.
- Compute the combination $\binom{n}{k}$. This shows the number of ways to get exactly k successes in n trials.
- Raise p to the power of k. This calculates the probability of those k successes.
- Raise (1 − p) to the power of (n − k). This calculates the probability of failure for the remaining trials.
- Multiply all parts together as follows: $\binom{n}{k} \times p^k \times (1-p)^{n-k}$ to get the exact probability of k successes.
Calculating Exact and Cumulative Binomial Probabilities: Solved Examples
Understanding binomial probability becomes easier when we look at real examples. Let’s use a simple but realistic scenario to show you to calculate both exact and cumulative probabilities of a binomial experiment.
Scenario: A light bulb manufacturer knows that about 3% of bulbs are defective. A quality inspector randomly tests 20 bulbs from a shipment. Let X be the number of defective bulbs.
From the scenario, we know that:
- n = 20 (number of trials)
- p = 0.03 (probability of a defective bulb — “success”)
- q = 1 – p = 0.97 (probability of a non-defective bulb — “failure”)
From the information, let’s see how you can easily calculate exact and cumulative probabilities using these examples:
Example 1. Probability That Exactly 2 Bulbs Are Defective, P(X = 2)
This is an exact binomial probability problem. We need to find P(X = 2).
Using the binomial formula:
P(X=2) = $(20C2)(0.03)^2(0.97)^{18}$
$= 190 \times (0.03)^2 \times (0.97)^{18}$
= 0.098830
Thus, the probability that exactly 2 bulbs are defective is 0.0988.
Example 2. Probability That at Most 2 Bulbs Are Defective, P(X ≤ 2)
This is a cumulative binomial probability problem. The inspector needs to find P(X ≤ 2).
By definition, P(X ≤ 2) = p(x=0) + p(x=1) + p(x=2)
In other words, we need to use the binomial probability formula, solve each of the exact probabilities, and sum them as follows:
Compute exact probability for each term:
P(X=0) = $(20C0)(0.03)^0(0.97)^{20}$
= 0.543794
P(X=1) = $(20C1)(0.03)^1(0.97)^{19}$
= 0.336368
P(X=2) = $(20C2)(0.03)^2(0.97)^{18}$
= 0.098830
Thus, P(X≤2)=0.543794+0.336368+0.098830
=0.978992
Therefore, the probability that at most 2 bulbs are defective is 0.9790.
Example 3. Probability That at Least 3 Bulbs Are Defective, P(X ≥ 3)
This is another cumulative binomial probability problem. “At least 3” means 3 or more defective bulbs. In this case, instead of calculating all exact probabilities from x =3 to x =20, we use the complement rule as follows:
P(X≥3)=1–P(X≤2)
But from example 2, P(X≤2)=0.978992
Thus, P(X≥3)=1– 0.978992
= 0.021008
Thus, the probability that at least 3 bulbs are defective is 0.0210.
Example 4. Probability That Less Than 2 Bulbs Are Defective, P(X < 2)
This is a cumulative binomial probability problem. Probability that x is “Less than 2” means x = 0 or 1 defective bulb.
In other words, P(X<2) is equivalent to P(X≤1)
Thus, we can define: P(X<2) =P(X=0) + P(X=1)
But from example 2, P(X=0)=0.543794 and P(X=1)=0.336368
Hence, P(X<2) = 0.543794 + 0.336368
= 0.880162
Therefore, the probability that less than 2 bulbs are defective is 0.8802.
Example 5. Probability That More Than 1 Bulb Is Defective, P(X > 1)
This is also a cumulative binomial probability problem. We can use the complement rule to find it. Thus, instead of computing all exact probabilities from x = 1 to x = 20, we only need to find 1-P(X≤1).
That’s, P(X>1)=1–P(X≤1)
From Example 4: P(X≤1)=0.880162
Hence, P(X>1)=1–0.880162
=0.119838
Therefore, the probability that more than 1 bulb is defective is 0.1198.
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Conclusion
Calculating binomial probability is not as hard as it first seems. Once you understand the formula and the meaning of n, p, and k, you can solve any problem step by step. The key is to identify whether the question asks for an exact or a cumulative probability.
In real situations, binomial probability helps make informed decisions; from testing product quality to predicting outcomes in research or marketing. It is one of the most useful tools in statistics for analyzing success-or-failure scenarios.
Frequently Asked Questions
Binomial probability measures the chance of getting a specific number of successes in a fixed number of trials. Each trial has only two possible outcomes: success or failure, and the probability of success remains constant.
You calculate binomial probability using the formula: P(X=k)=nCk * p^k * (1−p)^(n−k).
Where n is the number of trials, k is the number of successes, and p is the probability of success. You can also use a binomial probability calculator for faster results.
Exact probability is the chance of getting exactly k successes. On the other hand, Cumulative probability is the chance of getting up to or more than a certain number of successes (e.g., P(X ≤ 2) or P(X ≥ 3).
Use binomial probability when an experiment has:
– A fixed number of trials (n)
– Two possible outcomes per trial (success/failure)
– Constant probability of success (p)
– Independent trials
