A-level · Statistical distributions

Cumulative binomial probabilitiesExam-style questions

Finding P(X ≤ r), P(X < r), P(X ≥ r), P(X > r) and P(a ≤ X ≤ b) for a binomial distribution by building everything from the cumulative function P(X ≤ r) (binomcdf), with care over the off-by-one when an inequality is strict or "at least". Includes the complement for "≥" and "two-stage" problems where the success of one binomial becomes the trial of another.

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

DifficultyAll Level 3 Level 4 Level 5

1
A theatre has found that, over a long period, $30\%$ $30%$ of the people who attend a performance buy a programme.

A random sample of $25$ $25$ people who attend a performance is taken.

The random variable $X$ $X$ represents the number of these people who buy a programme.
(a) Using a suitable distribution for $X$ $X$ , find

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

(ii) $\mathrm{P}(X \leqslant 5)$ $P (X ⩽ 5)$

(3 marks)

The price of a programme is then increased.

The manager of the theatre claims that the proportion of people attending a performance who buy a programme has fallen.

After the increase, a random sample of $50$ $50$ people who attend a performance is taken and $9$ $9$ of them buy a programme.
(b) Use a suitable test to assess the manager's claim.

You should
- state your hypotheses clearly
- use a $5\%$ level of significance
(4 marks)
Level 4 · 7 marksTry it
2
In a certain country, 38% of people have blood group A. During one month, 1400 people give blood at a regional donor centre, and these donors may be regarded as a random sample from the population of the country. Let $X$ be the number of these donors who have blood group A.

Determine the largest integer $k$ for which $\mathrm{P}(X \leqslant k) < 0.01$ .

(5 marks)
Level 4 · 5 marksTry it