S6.2StatisticsStretch

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 — like counting heads in 10 coin flips. It is one of the most useful models in the course.

30 min Video by Zeeshan Zamurred Statistical Distributions

Edexcel AS Level Maths: 6.2 The Binomial DistributionWatch the full walkthrough before the notes below.

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What you'll be able to do

Recognise when a binomial model applies
State the distribution X ~ B(n, p)
Use the binomial probability formula
Calculate P(X = r)

When to use it

A binomial model fits when there are a $fixed$ number $n$ of $independent$ trials, each with only $two$ outcomes (success/failure) and the $same$ probability $p$ of success. We write $X \sim B (n, p)$ .

X \sim B (n, p)

n

trials, probability

p

of success each.

The probability formula

The probability of exactly $r$ successes uses a binomial coefficient (the “choose” function from the Pure binomial expansion).

P (X = r) = (r n) p^{r} (1 - p)^{n - r}

Choose which trials succeed, then their probabilities.

(2 5) = 10

10 \times 0. 5^{2} \times 0. 5^{3} = 10 \times 0.03125

Answer

0.3125

Tip — The exponents must add to n: pʳ has r, (1−p) has n−r.

Conditions check

Always confirm the four conditions before modelling with a binomial: fixed n, independent trials, two outcomes, constant p. If any fails (e.g. without replacement changes p), the binomial does not apply.

Formula recap

X \sim B (n, p)

Binomial notation.

P (X = r) = (r n) p^{r} (1 - p)^{n - r}

Probability of r successes.

fixed n, independent, 2 outcomes, constant p

The four conditions.

Common mistakes to avoid

Using a binomial when p changes between trials (e.g. without replacement).

p must stay constant and trials independent for a binomial.

Forgetting the binomial coefficient nCr in the formula.

P(X = r) includes the nCr factor for the number of arrangements.

Key takeaways

Binomial X ~ B(n, p): fixed n independent trials, constant p, two outcomes.
P(X = r) = nCr · pʳ · (1−p)ⁿ⁻ʳ.
Check the four conditions before using it.

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