A-level · Statistical distributions

The binomial distribution as a modelExam-style questions

When the count of "successes" in a fixed number of repeated trials follows a binomial distribution X ~ B(n, p): the four conditions (a fixed number of trials, two outcomes per trial, a constant probability of success, and independent trials), how to identify n and p from a context, how to state a condition for a binomial model, and when the binomial is NOT appropriate.

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18 questions · A-level level · mark scheme and worked solution on each · no sign-up to try

DifficultyAll Level 2 Level 3 Level 4

1
A bank uses a test to decide which new members of staff need training in spotting forged banknotes. The test has $30$ $30$ sets of notes. Each set contains $4$ $4$ banknotes, exactly one of which is forged, and the person taking the test must pick out the forged note.
- Each set in which the forged note is picked out scores $5$ points.
- Each set in which a genuine note is picked loses $2$ points.
Oskar is a new member of staff. He cannot tell forged notes from genuine ones, so he picks one note at random from each set.

Let $X$ $X$ be the number of sets in which Oskar picks out the forged note.
(a) State the distribution of $X$ $X$

(1 mark)

Let $T$ $T$ be Oskar's total score for the test.
(b) (i) Write down $T$ $T$ in terms of $X$ $X$

(ii) Hence, show clearly that the expected value of $T$ $T$ is $-7.5$ $- 7.5$

(4 marks)

A new member of staff who scores $20$ $20$ points or more does not need the training.
(c) Find the probability that Oskar does not need the training.

(3 marks)

Staff who have completed the training pick out the forged note in a set with probability $0.45$ $0.45$

Forty trained members of staff are chosen at random, and each of them attempts one set of notes. Let $Y$ $Y$ be the number of these $40$ $40$ staff who pick out the forged note.

Given that $\mathrm{P}(n < Y \leqslant 22) = 0.7907$ $P (n < Y ⩽ 22) = 0.7907$ to $4$ $4$ decimal places,
(d) find the value of $n$ $n$

(4 marks)
Level 4 · 12 marksTry it
2
A garden centre sells poppy seeds in small packets and in large packets. Each seed grows into a plant with red flowers with probability $p$ , independently of the other seeds.

A small packet holds $n$ seeds. For a small packet, the number of seeds that grow into plants with red flowers has mean $4.5$ and variance $3.15$
(a) Find the value of $n$ and the value of $p$

(3 marks)

A large packet holds $25$ seeds. The random variable $R$ represents the number of seeds in a large packet that grow into plants with red flowers.
(b) Find the probability that $R$ is greater than its mean.

(3 marks)
(c) Find the largest value of $r$ such that $\mathrm{P}(R < r) < 0.05$

(2 marks)

Jamal buys $6$ large packets.
(d) Find the probability that, for at least $5$ of these $6$ packets, the number of seeds that grow into plants with red flowers is greater than the mean of $R$

(3 marks)
Level 4 · 11 marksTry it
3
A machine produces components, of which $8\%$ are faulty. A random sample of $25$ components is taken, and $X$ is the number of faulty components in the sample.
(a) State two conditions needed for $X$ to be modelled by a binomial distribution. (2 marks)
(b) Find $\mathrm{P}(X \leqslant 3)$ , giving your answer to $4$ decimal places. (2 marks)
(c) Find $\mathrm{P}(X \geqslant 2)$ , giving your answer to $4$ decimal places. (2 marks)

The machine is serviced. A new random sample of $25$ components is taken, and none of them is faulty.
(d) Test, at the $5\%$ level of significance, whether the proportion of faulty components has decreased. (4 marks)
Level 4 · 10 marksTry it
4
A bag contains a large number of beads, $30\%$ of which are red. A handful of $12$ is taken. Explain why the number of red beads can be modelled approximately by a binomial distribution, despite the sampling being without replacement.
Level 4 · 1 markTry it
5
$2\%$ of items produced by a machine are faulty. A sample of $30$ items is taken. Suggest a model for the number of faulty items, and find the probability that none are faulty, to 4 d.p.
Level 4 · 3 marksTry it

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