CAIE A-Level Mathematics 5.4 Discrete Random Variables Question Bank

CAIE A-Level Mathematics 5.4 Discrete Random Variables Question Bank
Cambridge International AS & A Level Mathematics 9709 syllabus for exams in 2028, 2029 and 20302028–2030

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 variables5.4.2—Geometric distribution5.4.3—Expectation and variance

Question 1

[Maximum number: 4]

The random variable X takes the value x with probability kx2k x^{2}, where k is a constant and x takes the values -2,1,2,3 only.

Question 1(a)

(a)

Draw up the probability distribution table for X, giving the probabilities as numerical fractions.

[ 3 ]

Question 1(b)

(b)

Find E(X).

[ 1 ]

Question 1

[Maximum number: 3]

A coin is biased so that the probability of obtaining a head when it is thrown is 0.4. The coin is thrown repeatedly until the first head is obtained.

Question 1(a)

(a)

Find the probability that the first head is obtained on the 5th throw.

[ 1 ]

Question 1(b)

(b)

Find the probability that the first head is obtained after the 6th throw.

[ 2 ]

Question 1

[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.

Question 1(a)

(a)

State the value of E(X).

[ 1 ]

Question 1(b)

(b)

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

[ 1 ]

Question 1(c)

(c)

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

[ 2 ]