The Binomial Distribution
“Binomial = (success pᵏ) × (failure) × (placement count ₙCₖ).”
The formula
P(X=k) = ₙCₖ pᵏ (1−p)ⁿ⁻ᵏHow to read it: the probability of exactly k successes in n independent trials, each with success probability p
- n
- — the number of independent trials
- p
- — the probability of success in one trial
- k
- — the number of successes (0 to n)
- ₙCₖ
- — the number of ways to place k successes among n trials
The hook
What's the chance of exactly 6 heads in 10 coin flips? If it's just success-or-failure repeated, one formula gives the answer.
In plain words
It's the distribution of the probability of getting k successes when you run n independent trials, each with the same success probability p.
The intuition
It's a picture of three factors multiplied — the probability of k successes pᵏ, the probability of n−k failures (1−p)ⁿ⁻ᵏ, and the number of orders the successes can fall in, ₙCₖ. Ignore the ordering and you'd miss probability, so you count the placements with a combination and multiply.
How it's built
One specific order (say success-success-failure…) has probability pᵏ(1−p)ⁿ⁻ᵏ. There are ₙCₖ such orders, so summing them attaches the coefficient ₙCₖ. This distribution's mean is np — 'number of trials × success probability'.
Example
Three coin flips (p=1/2), probability of 2 heads: ₃C₂·(1/2)²·(1/2)¹ = 3·(1/8) = 3/8. There are 3 slots for the heads, hence the coefficient 3.
Common mistake
The binomial distribution applies only when the trials are 'independent' and the success probability p is 'constant'. Drawing without replacement changes the probability each time, so it is not binomial.
Where it's used
Defect counts in quality control, the number of yes-voters in a survey, responders in a clinical trial, predicting win totals in sports — any statistic that 'repeats success/failure'.
Where it came from
The name for repeated success/failure trials (Bernoulli trials) comes from Jacob Bernoulli in the 1600s, and together with his 'law of large numbers' it became a cornerstone of probability.
Prerequisites
Quick check
What is the probability of 2 heads in 2 coin flips? (p=1/2)
- 1/4✓
- 1/2
- 1/8
- 3/8
Practice
Find the probability of exactly 2 heads in 3 coin flips, as a decimal. (p=1/2)
Answer: 0.375
- ₃C₂·(1/2)²·(1/2)¹ = 3·(1/8)
- = 3/8 = 0.375
Takeaway: Don't forget the placement count ₃C₂=3.
Find the mean np of a binomial distribution with n=5, p=0.5.
Answer: 2.5
- Mean = np = 5·0.5
- = 2.5
Takeaway: A binomial distribution's mean is always np.
For one die roll, find the probability of rolling a 6 (success), as a decimal. (n=1, p=1/6)
Answer: 0.167
- ₁C₁·(1/6)¹·(5/6)⁰ = 1/6
- ≈ 0.167
Takeaway: With n=1 the binomial reduces to a single success probability.
Explain why P(X=0)+P(X=1)+…+P(X=n)=1.
Answer: undefined
- The number of successes must be one of 0 through n
- Adding every case leaves nothing out
- The total probability is always 1
Takeaway: The probabilities of all possible outcomes sum to 1.