The Binomial Distribution

This lesson covers the binomial distribution as required by the Edexcel A-Level Mathematics specification (9MA0), Paper 3 Section A -- Statistics. You need to understand the conditions for a binomial model, calculate probabilities using the formula, use cumulative probabilities, and know the mean and variance.

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Conditions for a Binomial Distribution

A random variable X follows a binomial distribution B(n, p) if all four of the following conditions are met:

1. There is 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.

Notation

X ~ B(n, p) where n = number of trials and p = probability of success on each trial.

Example

A fair coin is flipped 10 times. Let X = number of heads.

• n = 10 (fixed). Two outcomes per flip. P(heads) = 0.5 (constant). Flips are independent.

So X ~ B(10, 0.5).

Counter-example (not binomial)

Drawing 4 balls without replacement from a bag of 5 red and 3 blue -- the probability changes with each draw, so it is not binomial.