The Binomial Distribution

Fixed trials, two outcomes, constant probability. How C(n, x) p^x q^(n-x) gives the chance of exactly x successes, why the coefficient counts the possible orders, and what the expected value np tells you.

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A binomial experiment repeats the same trial a fixed number of times, where each trial has only two outcomes — success or failure. The binomial distribution then gives the probability of getting exactly successes.

Conditions When is it binomial?
  1. Two outcomes: each trial ends in success or failure.
  2. Fixed : the number of trials is known in advance.
  3. Constant : the success probability is the same on every trial.
  4. Independence: the result of one trial does not affect the others.

Typical examples: tossing a coin times ; inspecting items on a production line ; answering true/false questions .

Formula The binomial probability

The probability of exactly successes in trials is:

= number of trials, = number of successes wanted.
= probability of success, = probability of failure.
counts the ways to place successes among trials.
Example A coin tossed 6 times
31.25% 0 1 2 3 4 5 6 x = number of heads

Exactly 3 heads, with , , :

.
⟹ P(X = 3) = 31.25%.
Example Two more

A die rolled 10 times — exactly two sixes

Here , , .
.
⟹ P(X = 2) ≈ 29.1%.

A production line (5% defective) — inspect 20, exactly one defective

Here , , .
.
⟹ P(X = 1) ≈ 37.7%.
Note Expected number of successes

On average, a binomial experiment gives successes. For the coin above, — which is exactly where the distribution peaks.

Reference The three factors
Part Symbol Meaning
Combinations C(n, x) ways to place x successes in n trials
Success probability pˣ x successes occurring
Failure probability qⁿ⁻ˣ (n − x) failures occurring
Expected value μ = np average number of successes
Summary
  1. Conditions: two outcomes, a fixed , a constant , and independent trials.
  2. The law: — three factors multiplied together.
  3. is the number of ways to spread successes across trials.
  4. Expected value: — a larger or raises the average number of successes.
283. Relations and Functions
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