Binomial Distribution
The Binomial Distribution
The binomial distribution models the number of successes in a fixed number of independent trials, each with the same probability of success. It is one of the most important discrete distributions at A-Level.
Conditions for a Binomial Distribution
A random variable X follows a binomial distribution, written X ~ B(n, p), when:
1. There are a fixed number of trials, n
2. Each trial has exactly two outcomes (success or failure)
3. The probability of success, p, is constant for each trial
4. The trials are independent of each other
If any condition is violated, the binomial model is not appropriate. Common violations include sampling without replacement (which creates dependence) or changing probability (as in a sequence of increasingly difficult questions).
The Probability Formula
The probability of exactly r successes in n trials is:
P(X = r) = ⁿCᵣ × pʳ × (1 - p)ⁿ⁻ʳ
where:
- ⁿCᵣ = n! / (r!(n-r)!) counts the number of ways to choose which r trials are successes
- pʳ is the probability of r successes
- (1-p)ⁿ⁻ʳ is the probability of (n-r) failures
Worked Example: A fair die is rolled 10 times. Find the probability of getting exactly 3 sixes.
X = number of sixes. X ~ B(10, 1/6).
P(X = 3) = ¹⁰C₃ × (1/6)³ × (5/6)⁷
= 120 × (1/216) × (78125/279936)
= 120 × 78125 / 60466176
= 120 × 0.001292
≈ 0.1550
Cumulative Probabilities
Many questions require cumulative probabilities:
- P(X ≤ k) = P(X = 0) + P(X = 1) + ... + P(X = k) — use cumulative binomial tables or calculator
- P(X ≥ k) = 1 - P(X ≤ k - 1)
- P(X < k) = P(X ≤ k - 1)
- P(X > k) = 1 - P(X ≤ k)
- P(a ≤ X ≤ b) = P(X ≤ b) - P(X ≤ a - 1)
Worked Example: If X ~ B(20, 0.3), find P(X ≥ 8).
P(X ≥ 8) = 1 - P(X ≤ 7)
Using tables or calculator: P(X ≤ 7) = 0.7723
P(X ≥ 8) = 1 - 0.7723 = 0.2277
Mean, Variance and Standard Deviation
For X ~ B(n, p):
- Mean (expected value): E(X) = np
- Variance: Var(X) = np(1 - p) = npq (where q = 1 - p)
- Standard deviation: σ = √(np(1 - p))
Worked Example: In 200 trials with p = 0.35, find the mean and standard deviation.
Mean = 200 × 0.35 = 70
Variance = 200 × 0.35 × 0.65 = 45.5
Standard deviation = √45.5 ≈ 6.75
Modelling with the Binomial Distribution
Real-world applications include:
- Quality control (number of defective items in a batch)
- Medical trials (number of patients responding to treatment)
- Genetics (number of offspring with a particular trait)
- Surveys (number of people answering "yes")
When modelling, you must justify the conditions. For example, "each component is tested independently and the defect rate is constant at 2%" shows independence and constant p.
Worked Example: A factory produces bolts with a 5% defect rate. A sample of 15 bolts is taken. Find the probability that (a) exactly 2 are defective, (b) at most 1 is defective.
X = number of defective bolts. X ~ B(15, 0.05).
(a) P(X = 2) = ¹⁵C₂ × 0.05² × 0.95¹³ = 105 × 0.0025 × 0.5133 ≈ 0.1348
(b) P(X ≤ 1) = P(X = 0) + P(X = 1)
P(X = 0) = 0.95¹⁵ = 0.4633
P(X = 1) = 15 × 0.05 × 0.95¹⁴ = 15 × 0.05 × 0.4877 = 0.3658
P(X ≤ 1) = 0.4633 + 0.3658 = 0.8290
When to Use Other Distributions
- If n is large and p is very small, use the Poisson distribution as an approximation
- If n is large and np > 5, np(1-p) > 5, consider the normal approximation to the binomial (with continuity correction)
Exam Tips
- Always define your random variable (e.g., "Let X be the number of...") and state the distribution (X ~ B(n, p)).
- For cumulative probabilities, be precise about inequalities: P(X ≥ 5) = 1 - P(X ≤ 4), not 1 - P(X ≤ 5).
- Use tables or a calculator for cumulative binomial probabilities — computing each term individually is slow and error-prone.
- When checking if the binomial model is appropriate, address all four conditions explicitly.
- Remember: ⁿCᵣ counts combinations (order does not matter). SSSR is different from RSSS as a sequence but not as a count.