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AS Statistics · Edexcel 8MA0 · free A-level maths worksheet

The binomial distribution

Conditions for a binomial model, calculating P(X = r), P(X <= r) and P(X >= r), tables and calculators.

12 original questions, 62 marks. Print the worksheet, mark it with the answers sheet, and use the solutions to see every step written out by hand.

Worksheet PDFWorked solutions PDFAnswers PDF

The questions

1

A factory packs light bulbs in boxes of 20. The random variable X is the number of faulty bulbs in a box.

  1. State two conditions needed for X to be modelled by a binomial distribution.
  2. Sam deals 5 cards, without replacement, from a standard pack of 52 playing cards. The random variable H is the number of hearts dealt. Explain why H cannot be modelled by B(5, 0.25).
3 marks
Show answer
a) any two of: each bulb is faulty or not faulty; bulbs are faulty independently of each other; the probability of a bulb being faulty is the same for every bulb; a fixed number (20) of bulbs b) without replacement, the probability of a heart changes from card to card (the trials are not independent)
2

The random variable X ~ B(12, 0.3). Find

  1. P(X = 4)
  2. P(X ≤ 3)
  3. P(X > 5)

giving each answer to 4 decimal places.

3 marks
Show answer
a) 0.2311 b) 0.4925 c) 0.1178
3

The random variable X ~ B(20, 0.4). Find, to 4 decimal places,

  1. P(X ≥ 10)
  2. P(5 < X ≤ 9)
4 marks
Show answer
a) 0.2447 b) 0.6297
4

A café finds that 15% of its customers order a drink with oat milk. On one morning a random sample of 25 customers is taken. The random variable X is the number of these customers who order a drink with oat milk.

  1. Write down a suitable distribution to model X.
  2. Find the probability that exactly 4 of the customers order a drink with oat milk.
  3. Find the probability that at most 2 of the customers order a drink with oat milk.
  4. Find the probability that more than 5 of the customers order a drink with oat milk.
5 marks
Show answer
a) X ~ B(25, 0.15) b) 0.2110 c) 0.2537 d) 0.1615
5

In a town, 45% of learner drivers pass their driving test at the first attempt. A random sample of 10 learner drivers from the town is taken.

  1. Find the probability that exactly 6 of them pass at the first attempt.
  2. Find the probability that at least 4 of them pass at the first attempt.
  3. Priya says the same model could be used for 10 friends who all took lessons with the same instructor. Give one reason why the model may not be suitable for them.
5 marks
Show answer
a) 0.1596 b) 0.7340 c) the friends had the same instructor, so their chances of passing are probably not 0.45 and may not be independent
6

The random variable X ~ B(30, 0.25).

  1. Find P(4 ≤ X < 10).
  2. Find the largest value of k such that P(X ≤ k) < 0.1
  3. Find the smallest value of m such that P(X ≥ m) < 0.05
6 marks
Show answer
a) 0.7660 b) k = 4 c) m = 13
7

On any school day, the probability that a particular bus arrives late is 0.2, independently of other days.

  1. Find the probability that the bus is late on exactly 2 of the 5 days in one week.
  2. Find the smallest number of school days, n, for which the probability that the bus is late on at least one of the days is greater than 0.95
5 marks
Show answer
a) 0.2048 b) n = 14
8

Lily rolls a fair six-sided die 8 times and records X, the number of sixes.

  1. Find P(X = 2).
  2. Lily repeats this experiment once a day for 12 days. Find the probability that she gets exactly 2 sixes on at least 3 of the days.
6 marks
Show answer
a) 0.2605 b) 0.6411
9

A spinner lands on red with probability p on each spin. It is spun 5 times and X is the number of times it lands on red.

Given that P(X = 0) = 0.07776

  1. show that p = 0.4
  2. find P(X = 2)
  3. find P(X ≥ 4).
6 marks
Show answer
a) (1 - p)5 = 0.07776 so 1 - p = 0.6 and p = 0.4 b) 0.3456 c) 0.0870
10

A seed company sells packets of 30 sunflower seeds. Each seed germinates with probability 0.9, independently of the others.

  1. Find the probability that at least 25 seeds in a packet germinate.
  2. Kwame buys 6 packets. Find the probability that exactly 4 of his packets have at least 25 seeds that germinate.
6 marks
Show answer
a) 0.9268 b) 0.0593
11

The random variable X ~ B(n, 0.25). Given that P(X = 1) = P(X = 2)

  1. show that n = 7
  2. find P(X ≤ 1).
6 marks
Show answer
a) n(0.25)(0.75)(n - 1) = n(n - 1)/2 (0.25)2 (0.75)(n - 2) gives 0.75 = (n - 1)/8, so n = 7 b) 0.4449
12

Ana is an archer. Each arrow she shoots hits the gold ring with probability 0.35, independently of other arrows. She shoots arrows in rounds of 8 arrows, and X is the number of golds in a round.

  1. Find P(X ≥ 3).
  2. Find the probability that she gets exactly 2 golds in a round.
  3. Ana shoots 3 rounds. Find the probability that she gets at least 10 golds altogether.
  4. Find the probability that she gets at least 3 golds in each of the 3 rounds.
7 marks
Show answer
a) 0.5722 b) 0.2587 c) 0.3134 d) 0.1873

Every question here is original and written by Robinson Tuition. It follows the Edexcel AS specification and suits AQA and OCR AS maths too; this is not an exam-board paper.

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