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
Twenty seeds are planted; each germinates independently with probability $0.8$ . Suggest a suitable model for the number that germinate, and find the probability that all $20$ germinate, to 4 d.p.
Level 3 · 3 marksTry it
2
Five cards are drawn without replacement from a standard pack, and $X$ is the number of hearts. Explain why $X$ does not follow a binomial distribution.
Level 3 · 1 markTry it
3
A multiple-choice test has $15$ questions, each with $4$ options. A student guesses every answer. Let $X$ be the number correct.
(a) State the distribution of $X$ .
(b) Find $P(X = 5)$ , to 4 d.p.
Level 3 · 3 marksTry it
4
Over many years, $12\%$ of the tulip bulbs sold by a garden centre have failed to flower.

The garden centre sells these bulbs in bags of $25$

For a bag chosen at random, the number of bulbs in the bag that fail to flower is modelled by the distribution $\text{B}(25, 0.12)$
(a) State one assumption, in context, that is needed for this distribution to be used.

(1 mark)
(b) Find the probability that exactly $3$ of the bulbs in the bag fail to flower.

(1 mark)
(c) Find the probability that fewer than $5$ of the bulbs in the bag fail to flower.

(2 marks)
Level 3 · 4 marksTry it
5
Nadia is practising on a climbing wall.

Each time Nadia attempts a particular route, the probability that she completes the route without resting is $0.35$

Nadia attempts the route $18$ times.

The random variable $X$ represents the number of times Nadia completes the route without resting.
(a) State a condition needed for $X$ to be modelled by a binomial distribution.

(1 mark)
(b) Using a suitable distribution for $X$ , find

(i) $\mathrm{P}(X = 5)$

(ii) $\mathrm{P}(X \geqslant 8)$

(3 marks)
Level 3 · 4 marksTry it