Binomial Distribution
Learn the conditions for a binomial setting, the P(X = k) formula, and how to find the mean and variance of a binomial random variable.
Conditions for a Binomial Distribution
A random variable X follows a binomial distribution if the experiment satisfies all four of these conditions:
Fixed number of trials (n)
The experiment is repeated a set number of times.
Two outcomes per trial
Each trial results in either "success" or "failure."
Constant probability (p)
The probability of success is the same for every trial.
Independent trials
The outcome of one trial does not affect any other trial.
We write X ~ B(n, p) to indicate X follows a binomial distribution with n trials and success probability p.
The Binomial Probability Formula
The probability of getting exactly k successes in n trials is:
Binomial Probability
P(X = k) = nCk · pk · (1-p)n-k
nCk
Ways to arrange k successes among n trials
pk
Probability of k successes
(1-p)n-k
Probability of (n-k) failures
Mean and Variance of Binomial
For X ~ B(n, p), the mean, variance, and standard deviation have elegant formulas:
Mean
μ = np
Variance
σ2 = np(1-p)
Standard Deviation
σ = √[np(1-p)]
Key Vocabulary
Binomial Distribution
A distribution that models the number of successes in n independent trials, each with probability p.
Trial
A single repetition of the experiment, with exactly two possible outcomes.
Success Probability (p)
The constant probability of the desired outcome on each trial. q = 1 - p is the failure probability.
B(n, p)
Notation indicating X follows a binomial distribution with n trials and success probability p.
Worked Examples
A fair coin is tossed 5 times. Find P(exactly 3 heads).
Step 1: X ~ B(5, 0.5). We need P(X = 3).
Step 2: P(X = 3) = 5C3 × (0.5)3 × (0.5)2 = 10 × 0.125 × 0.25
Answer: P(X = 3) = 0.3125
X ~ B(20, 0.3). Find the mean and standard deviation.
Step 1: Mean = np = 20 × 0.3 = 6
Step 2: Variance = np(1-p) = 20 × 0.3 × 0.7 = 4.2
Answer: Mean = 6, SD = √4.2 ≈ 2.049
A multiple choice test has 8 questions, each with 4 options. If a student guesses randomly, find P(exactly 2 correct).
Step 1: X ~ B(8, 0.25). We need P(X = 2).
Step 2: P(X = 2) = 8C2 × (0.25)2 × (0.75)6 = 28 × 0.0625 × 0.177979
Answer: P(X = 2) ≈ 0.3115
Knowledge Check
Select the correct answer for each question. Click "Check Answer" to see if you are right.
Question 1
Which is NOT a condition for a binomial distribution?
Question 2
X ~ B(10, 0.4). What is E(X)?
Question 3
X ~ B(6, 0.5). What is P(X = 0)?
Question 4
X ~ B(10, 0.4). What is Var(X)?
Question 5
A factory produces items with a 10% defect rate. In a sample of 5 items, what is P(exactly 1 defective)?
Key Concepts Summary
- ●A binomial requires: fixed n, two outcomes, constant p, independent trials.
- ●P(X = k) = nCk pk (1-p)n-k.
- ●Mean: μ = np. Variance: σ2 = np(1-p).
- ●Standard deviation: σ = √[np(1-p)].
- ●The binomial connects directly to combinations (the nCk term).