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
$X \sim B(12, 0.25)$ . Find $P(X = 3)$ , to 4 d.p.
Level 2 · 2 marksTry it
2
$X \sim B(15, 0.4)$ . Find $P(X = 6)$ , to 4 d.p.
Level 2 · 2 marksTry it
3
A café runs a promotion in which every hot drink sold comes with a scratch card.

For each scratch card
- the card either wins a free pastry or does not win a free pastry
- cards win independently of each other
The probability that a scratch card wins a free pastry is $0.15$ $0.15$
(a) Find the probability that

(i) exactly $3$ $3$ of $12$ $12$ scratch cards win a free pastry,

(ii) at least $4$ $4$ of $30$ $30$ scratch cards win a free pastry.

(4 marks)

Maya buys one hot drink from the café on each of $n$ $n$ days.
(b) Find the minimum value of $n$ $n$ such that the probability that at least one of Maya's scratch cards wins a free pastry exceeds $0.9$ $0.9$

(4 marks)
Level 3 · 8 marksTry it
4
On any given day, the probability that a certain machine breaks down is $0.15$ , independently of all other days.

Find the probability that, in a period of $12$ days, the machine breaks down
(a) exactly $3$ times, (2 marks)
(b) at least $3$ times. (3 marks)
Level 3 · 5 marksTry it
5
$X \sim B(20, 0.15)$ . Find $P(X = 2)$ , to 4 d.p.
Level 3 · 2 marksTry it

11 more questions on calculating binomial probabilities p(x = r)

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