Discrete Random Variables: Mathematics 9709 (Cambridge International AS & A Level)

A fair spinner is divided into four sectors labelled $1$, $2$, $3$ and $4$. The random variable $X$ denotes the score obtained on one spin, and has the probability distribution shown in the table below, where $k$ is a constant.

What is the value of $k$?

A small class is surveyed, and the random variable $X$ represents the number of pets owned by a randomly chosen student from the class. The probability distribution of $X$ is shown in the table below, where $k$ is a constant.

(a) Find the value of $k$. [2]

(b) Find $E(X)$. [2]

(c) Find $\mathrm{Var}(X)$. [3]

A factory manufactures LED bulbs. From long-run production records, the probability that a randomly chosen bulb is defective is $0.08$, and whether or not one bulb is defective has no effect on any other bulb. A quality inspector selects a random sample of $10$ bulbs from the production line, and $X$ denotes the number of defective bulbs found in the sample.

(a) State two conditions, in the context of this sample, that must hold for $X$ to be modelled by a binomial distribution. [2]

(b) Find $P(X=2)$. [3]

(c) Find $P(X \geq 2)$. [4]

(d) Write down $E(X)$. [1]

A biased six-sided die is rolled $8$ times. On each roll, the probability of obtaining a six is $0.3$, independently of any other roll. Let $X$ be the number of sixes obtained in the $8$ rolls, so that $X\sim B(8,0.3)$.

What is $P(X=2)$, correct to $3$ significant figures?

At a school fair, a tombola stall gives out a token to every player. Each token shows a number of points, and the random variable $X$ represents the number of points on a token drawn at random, with possible values $0$, $1$, $3$ and $5$. The probability distribution of $X$ is shown in the table below, where $k$ is a constant.

(a) Find the value of $k$. [2]

(b) Find $E(X)$. [3]

An archer shoots repeatedly at a target. On each shot, independently of any other shot, the probability that she hits the bullseye is $0.2$. Let $X$ be the number of shots up to and including her first bullseye, so that $X\sim \mathrm{Geo}(0.2)$.

What is $P(X=4)$, correct to $3$ significant figures?

In a large batch of memory cards, the probability that a randomly chosen card is defective is $0.15$, independently of any other card. A quality inspector selects cards one at a time, at random from the batch, and tests them until she finds the first defective card. Let $X$ be the number of cards tested up to and including the first defective card found, so that $X\sim \mathrm{Geo}(0.15)$.

(a) State two conditions, in the context of this situation, that must hold for $X$ to be modelled by the geometric distribution $\mathrm{Geo}(p)$. [2]

(b) Find $P(X=5)$. [2]

(c) Find $P(X \leq 3)$. [3]

(d) State $E(X)$. [1]

A multiple-choice quiz has $12$ questions, each with $5$ possible answers of which exactly one is correct. A student who has not revised guesses the answer to every question at random, independently of every other question. Let $X$ be the number of questions the student answers correctly.

(a) State the values of $n$ and $p$, and explain why $X$ can be modelled by a binomial distribution in this context. [2]

(b) Find $P(X=4)$. [3]

(c) Find $E(X)$ and $\mathrm{Var}(X)$. [3]

A biased coin is tossed $20$ times. On each toss, the probability of obtaining heads is $0.35$, independently of any other toss. Let $X$ be the number of heads obtained in the $20$ tosses, so that $X\sim B(20,0.35)$.

What is $\mathrm{Var}(X)$?

In a mock driving test, the random variable $X$ represents the number of minor faults recorded for a randomly chosen candidate, where $X$ can take the values $0$, $1$, $2$ or $3$. The probability distribution of $X$ is shown in the table below, where $p$ is a constant.

(a) Find the value of $p$. [3]

(b) Find $E(X)$. [2]

(c) Find $\mathrm{Var}(X)$. [3]