A-level · Statistical distributions

Calculating binomial probabilities P(X = r)Exam-style questions

The binomial probability formula P(X = r) = nCr · p^r · (1−p)^(n−r), why it has that shape (choose which r of the n trials succeed, then multiply the success and failure probabilities), how to evaluate it including the special cases r = 0 and r = n, and how it matches the binompdf function on a calculator.

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

DifficultyAll Level 2 Level 3 Level 4 Level 5

1
A seed company claims that $85\%$ of its parsnip seeds germinate. A gardener plants the seeds in trays of $20$ , and the number of seeds in a tray that germinate is modelled by the distribution $\mathrm{B}(20,\ 0.85)$ .
(a) Find the probability that, in a randomly chosen tray, exactly $18$ seeds germinate. (2 marks)
(b) Find the probability that, in a randomly chosen tray, at least $17$ seeds germinate. (2 marks)

The gardener suspects that the company's claim overstates the germination rate. In the next tray planted, only $13$ of the $20$ seeds germinate.
(c) Test, at the $5\%$ significance level, whether there is evidence to support the gardener's suspicion. State your hypotheses clearly. (5 marks)
Level 4 · 9 marksTry it
2
At a fairground stall, each player pays to have one go at a game.

Each time a player has a go, the probability that the player wins a prize is $0.18$ , independently of all other goes.

Maya has $16$ goes at this game.

The random variable $X$ represents the number of prizes that Maya wins.
(a) Using a suitable distribution for $X$ , find

(i) $\mathrm{P}(X = 4)$

(ii) $\mathrm{P}(X < 3)$

(3 marks)

During one weekend there are $300$ goes at this stall.
(b) Use a normal approximation to estimate the probability that more than $62$ prizes are won during this weekend.

(3 marks)
Level 4 · 6 marksTry it
3
Grace plays netball. At the end of each training session, she takes $12$ shots at goal.

For each of these shots, the probability that Grace scores is $0.7$ , independently of every other shot.

The random variable $X$ represents the number of goals that Grace scores from the $12$ shots in a training session.
(a) Using a suitable distribution for $X$ , find

(i) $\mathrm{P}(X = 9)$

(ii) $\mathrm{P}(X < 7)$

(3 marks)

Grace goes to $30$ training sessions during the season.
(b) Find the probability that Grace scores fewer than $7$ goals from her $12$ shots in at least $5$ of these sessions.

(3 marks)
(c) Find an estimate for the total number of goals that Grace scores from these shots in the $30$ sessions.

(1 mark)
Level 4 · 7 marksTry it