The binomial distribution as a model

Conditions

X ~ B(n, p) is a suitable model when:

1. there is a fixed number of trials, n;

2. each trial has two outcomes (success or failure);

3. the trials are independent;

4. the probability of success, p, is the same for every trial.

X is the number of successes, taking the values 0, 1, ..., n.

The probability formula

P(X = r) = C(n, r) p^r (1 - p)^{n - r}, where C(n, r) = n! / (r!(n - r)!) is the number of ways to choose which r trials are successes. The formula is in the formulae booklet, but you must be able to use it quickly.

Commenting on a model

Exam questions ask you to criticise a binomial model in context. Ask: is the number of trials really fixed? Could one trial affect another (for example, a player's confidence after a miss)? Is p really constant (for example, items drawn without replacement from a small batch)? Always tie the comment to the situation in the question.

Worked example

A seed has probability 0.3 of growing into a flowering plant. 12 seeds are planted. Find the probability that exactly 4 flower.

1. Let X be the number that flower: X ~ B(12, 0.3), assuming seeds grow independently.
2. P(X = 4) = C(12, 4) × 0.3⁴ × 0.7⁸
3. = 495 × 0.0081 × 0.057648

Answer: P(X = 4) = 0.2311

Check yourself

1. X ~ B(10, 0.25). Find P(X = 3). Give your answer to 4 decimal places.
   Show answer

   0.2503. C(10, 3) × 0.25³ × 0.75⁷ = 120 × 0.015625 × 0.133484 = 0.2503.

2. The probability that a light bulb is faulty is 0.15. In a box of 8 bulbs, find the probability that none is faulty. Give your answer to 4 decimal places.
   Show answer

   0.2725. X ~ B(8, 0.15): P(X = 0) = 0.85⁸ = 0.2725.

3. Using the box of 8 bulbs above, find the probability that at least one is faulty. Give your answer to 4 decimal places.
   Show answer

   0.7275. 1 - P(X = 0) = 1 - 0.2725 = 0.7275.

4. X ~ B(6, 0.4). Find P(X = 2 or X = 3). Give your answer to 4 decimal places.
   Show answer

   0.5875. P(X = 2) = 15 × 0.16 × 0.1296 = 0.3110; P(X = 3) = 20 × 0.064 × 0.216 = 0.2765; total 0.5875.

5. A fair die is thrown 20 times. Which distribution models the number of sixes?
   • B(6, 1/20)
   • B(20, 1/6)
   • B(20, 1/2)
   • Po(20)
   Show answer

   B(20, 1/6). 20 independent throws, each with P(six) = 1/6.

6. Which of these is least suitable for a binomial model?
   • The number of heads when a coin is tossed 30 times
   • The number of correct answers when a student guesses all 10 four-option questions
   • The number of faulty items in a sample of 50 from a very large production run
   • The number of red cards when 5 cards are dealt from one ordinary pack without replacement
   Show answer

   The number of red cards when 5 cards are dealt from one ordinary pack without replacement. Without replacement from 52 cards, the probability of red changes after each card, so p is not constant and trials are not independent.

7. A basketball player takes 15 free throws and the number scored is modelled as B(15, 0.7). Which is the best criticism of the model in context?
   • The number of throws is not fixed
   • Missing a shot might affect her confidence, so the throws may not be independent
   • There are more than two outcomes for each throw
   • 0.7 is too large to be a probability
   Show answer

   Missing a shot might affect her confidence, so the throws may not be independent. A criticism must link to the context: independence or a constant p is doubtful for a person.

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Sources

• Pearson Edexcel International Advanced Level in Mathematics specification (issue 3), Unit S2

Last updated 8 November 2026. Spotted a mistake? Tell us.