Paper 2 Section B Practice: Special Discrete Random Variables

Original practice material

These core 9758 questions total 20 marks. Poisson-distribution enrichment is deliberately excluded from this exam-like set.

Answers: Special Discrete Random Variables Answers.

Question 1: modelling seed germination

A seed from a large batch germinates with probability $0.72$. Twelve seeds are selected at random. Let $X$ be the number that germinate.

(a) State two assumptions needed for $X$ to be modelled by a binomial distribution. [2]

(b) State the distribution of $X$. [1]

(c) Find

(i) $P(X=9)$, [2]

(ii) $P(X\ge 9)$. [2]

(d) Find $E(X)$ and $\mathrm{Var}(X)$. [2]

(e) Given that at least 9 seeds germinate, find the probability that exactly 9 germinate. [2]

Question 2: determining an unknown probability

For a different variety, the number $Y$ of seeds that germinate out of 20 is modelled by $B(20,p)$.

(a) Given that $E(Y)=6.4$, find $p$. [2]

(b) Find $P(4\le Y\le 8)$. [3]

(c) Packets each contain 20 seeds. Find the smallest integer $m$ such that the probability that at least one of $m$ independently selected packets contains 9 or more germinating seeds exceeds $0.95$. [4]