CAIE A-Level Mathematics AS 5.4 Discrete Random Variables Questions

Practise constructing discrete probability distributions, calculating expectation and variance and selecting binomial or geometric models with parameters tied to the experiment.

Exam points

list every possible value of X with probabilities that are non-negative and sum to one
calculate E(X), E(X²) and Var(X) from a distribution table or stated formula
identify binomial or geometric conditions and state the model with its correct parameters

Covered syllabus points

5.4.1—Discrete random variables 5.4.2—Geometric distribution 5.4.3—Expectation and variance

[Maximum number: 6]

The numbers on the faces of a fair six-sided dice are 1,2,2,3,3,3. The random variable X is the total score when the dice is rolled twice.

(a)

Draw up the probability distribution table for X.

[ 3 ]

The probability distribution is

x	2	3	4	5	6
P(X=x)	$\frac{1}{36}$	$\frac{4}{36}=\frac19$	$\frac{10}{36}=\frac5{18}$	$\frac{12}{36}=\frac13$	$\frac{9}{36}=\frac14$

(b)

Find the value of $\operatorname{Var}(X)$ .

[ 3 ]

The mean is
$E(X)=\frac{1(2)+4(3)+10(4)+12(5)+9(6)}{36}=\frac{14}{3}.$
Therefore,
$\begin{aligned} \operatorname{Var}(X) &=\frac{1(2^2)+4(3^2)+10(4^2)+12(5^2)+9(6^2)}{36}-\left(\frac{14}{3}\right)^2\\ &=\frac{10}{9}. \end{aligned}$

[Maximum number: 6]

Salah decides to attempt the crossword puzzle in his newspaper each day. The probability that he will complete the puzzle on any given day is 0.65 , independent of other days.

(a)

Find the probability that Salah completes the puzzle for the first time on the 5th day.

[ 1 ]

To complete the puzzle for the first time on day 5, Salah must fail on the first four days and succeed on day 5:
$P=0.35^4(0.65)=0.00975.$

(b)

Find the probability that Salah completes the puzzle for the second time on the 5th day.

[ 2 ]

For the second success to occur on day 5, there must be exactly one success in the first four days, followed by a success on day 5:
$P=\binom41(0.35)^3(0.65)^2=0.0725.$

(c)

Find the probability that Salah completes the puzzle fewer than 5 times in a week ( 7 days).

[ 3 ]

Let X be the number of completed puzzles in 7 days. Then $X\sim\operatorname{Bin}(7,0.65)$ . Hence,
$\begin{aligned} P(X<5) &=1-P(X\geq5)\\ &=1-\left[\binom75(0.65)^5(0.35)^2 +\binom76(0.65)^6(0.35)+0.65^7\right]\\ &=0.468. \end{aligned}$

[Maximum number: 4]

Two fair coins are thrown at the same time repeatedly until a pair of heads is obtained. The number of throws taken is denoted by the random variable X.

(a)

State the value of E(X).

[ 1 ]

The probability of obtaining a pair of heads on one throw is 1/4. Therefore,
$E(X)=\frac{1}{1/4}=4.$

(b)

Find the probability that exactly 5 throws are required to obtain a pair of heads.

[ 1 ]

To require exactly five throws, the first four throws must not be two heads and the fifth must be two heads:
$P(X=5)=\left(\frac34\right)^4\left(\frac14\right)=\frac{81}{1024}=0.0791.$

(c)

Find the probability that fewer than 7 throws are required to obtain a pair of heads.

[ 2 ]

The probability of obtaining two heads in fewer than seven throws is
$P(X<7)=1-\left(\frac34\right)^6=\frac{3367}{4096}=0.822.$

All question bank results loaded

Covered syllabus points

Question 1

Question (a)

Question (b)

Question 2

Question (a)

Question (b)

Question (c)

Question 3

Question (a)

Question (b)

Question (c)