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 manufacturer claims that a treatment is effective for $60\%$ of patients. In a trial, the treatment is given to a random sample of $20$ patients, and $16$ of them improve.

A researcher wishes to test, at the $5\%$ level of significance, whether the proportion of patients for whom the treatment is effective is greater than $0.6$ .
(a) Carry out the test, stating your hypotheses clearly. (5 marks)
(b) Find the critical region for this test. (4 marks)
Level 5 · 9 marksTry it
2
Dev works for a company that sells home insurance over the telephone.

Each call that Dev makes leads to a sale with probability $0.16$ , independently of all his other calls.

On every working day, Dev makes $20$ calls.

The random variable $X$ represents the number of sales that Dev makes on one working day.
(a) Using a suitable distribution for $X$ , find

(i) $\mathrm{P}(X = 3)$

(ii) $\mathrm{P}(X \geqslant 5)$

(3 marks)

Dev's manager will review Dev's sales over the next $25$ working days.
(b) Find the probability that Dev makes exactly $3$ sales on at least $8$ of these $25$ days.

(3 marks)
(c) Find an estimate for Dev's total number of sales in the $25$ days.

(1 mark)

In a later month, Dev makes $400$ calls.
(d) Use a normal approximation to estimate the probability that more than $70$ of these calls lead to a sale.

(4 marks)
Level 5 · 11 marksTry it
3
In this question you must show detailed reasoning.

In a large population of a species of beetle, 28% of the beetles have a red spot on their backs. Beetles have the red spot independently of one another.
(a) An ecologist collects 500 of these beetles at random. The number of beetles in her collection that have a red spot is denoted by $Y$ . Determine the smallest integer $a$ for which $\mathrm{P}(Y \geqslant a) < 0.05$ .

(3 marks)

The binomial expansion of $(0.28 + 0.72)^{30}$ is written as $T_0 + T_1 + T_2 + \cdots + T_{30}$ , where $T_r$ is the term that contains $0.28^r$ .
(b) Show that $\dfrac{T_{r+1}}{T_r} = \dfrac{7(30 - r)}{18(r + 1)}$ .

(3 marks)

A student collects 30 of these beetles at random. The number of these beetles that have a red spot is modelled by the random variable $X$ .
(c) (i) Find the set of values of $r$ for which $\mathrm{P}(X = r + 1) \geqslant \mathrm{P}(X = r)$ .

(ii) Hence determine the most likely number of beetles with a red spot among the student's 30 beetles.

(6 marks)
Level 5 · 12 marksTry it