Binomial Distribution
Probability of k successes in n trials.
Formula
Variables
Example
Flipping 10 fair coins, P(exactly 5 heads) = C(10,5)·0.5¹⁰ ≈ 0.246.
How it works
The binomial formula gives the probability of getting exactly k successes in n independent trials when each trial has only two outcomes and the success probability stays constant at p. It works by multiplying the probability of one specific success/failure pattern, p^k(1-p)^(n-k), by the number of different ways those k successes can be arranged among the n trials, which is the combination term C(n,k). This counts every distinct sequence that contains the same number of successes exactly once.
When to use it
Use this when you are counting exact numbers of successes in repeated independent trials with the same success chance each time, such as defect-free items in quality control, correct answers on multiple-choice guesses, or the number of patients who respond to a treatment. It is appropriate when each trial is binary, independent, and the probability of success does not change from trial to trial.
Worked example, step by step
A basketball player has a 70% chance of making each free throw, independently. Find the probability of making exactly 8 out of 12 shots. Step 1: Identify the values: n = 12, k = 8, p = 0.7. Step 2: Substitute into the binomial formula: P(X = 8) = C(12,8)(0.7)^8(1 - 0.7)^(12 - 8) Step 3: Simplify the complement: P(X = 8) = C(12,8)(0.7)^8(0.3)^4 Step 4: Evaluate the combination: C(12,8) = 495 Step 5: Compute the powers: (0.7)^8 = 0.05764801 (0.3)^4 = 0.0081 Step 6: Multiply: P(X = 8) = 495 × 0.05764801 × 0.0081 P(X = 8) ≈ 0.2311 Final answer: P(X = 8) ≈ 0.231, so there is about a 23.1% chance of making exactly 8 of the 12 shots.
Common mistakes to avoid
- Using the formula when trials are not independent or the success probability changes from trial to trial
- Forgetting the combination factor C(n,k), which undercounts the different success arrangements
- Mixing up k and n-k, especially when writing the powers of p and 1-p
Did You Know?
The binomial distribution models coin flips and quality control — any fixed number of yes/no trials.
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