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AS Statistics · Edexcel 8MA0 · free A-level maths worksheet

Discrete random variables

Probability distributions, finding unknown probabilities, the discrete uniform distribution, cumulative probabilities.

12 original questions, 62 marks. Print the worksheet, mark it with the answers sheet, and use the solutions to see every step written out by hand.

Worksheet PDFWorked solutions PDFAnswers PDF

The questions

1

The discrete random variable X has the probability distribution

x1234
P(X = x)0.150.3k0.2
  1. Find the value of k.
  2. Find P(X ≥ 2).
  3. Find P(1 < X < 4).
3 marks
Show answer
a) k = 0.35 b) 0.85 c) 0.65
2

A fair spinner has eight equal sections numbered 1 to 8. The random variable X is the number the spinner lands on.

  1. Write down the probability distribution of X.
  2. Find P(X > 5).
  3. Find the probability that X is a prime number.
3 marks
Show answer
a) P(X = x) = 1/8 for x = 1, 2, ..., 8 (discrete uniform) b) 3/8 c) 1/2
3

The discrete random variable X has probability function

P(X = x) = kx    for x = 1, 2, 3, 4, 5

where k is a constant.

  1. Find the value of k.
  2. Find P(X = 4).
  3. Find P(X ≤ 2).
4 marks
Show answer
a) k = 1/15 b) 4/15 c) 1/5
4

The discrete random variable X has the probability distribution

x01234
P(X = x)0.1a0.25b0.15

where a and b are constants. Given that P(X ≤ 1) = P(X ≥ 3)

  1. find the value of a and the value of b
  2. find P(1 ≤ X < 4).
5 marks
Show answer
a) a = 0.275, b = 0.225 b) 0.75
5

The discrete random variable X takes the values 1, 2, 3, 4 and 5 only. Its cumulative distribution function F(x) = P(X ≤ x) is given in the table.

x12345
F(x)0.20.450.70.91
  1. Find P(X = 3).
  2. Find the probability distribution of X.
  3. Find P(X > 2).
5 marks
Show answer
a) 0.25 b) P(X = 1, 2, 3, 4, 5) = 0.2, 0.25, 0.25, 0.2, 0.1 c) 0.55
6

The discrete random variable X has probability function

P(X = x) = k(x2 + 1)    for x = 0, 1, 2, 3

  1. Show that k = 118
  2. Find P(X ≥ 2).
  3. Two independent observations X1 and X2 are taken of X. Find P(X1 + X2 = 3).
6 marks
Show answer
a) k(1 + 2 + 5 + 10) = 18k = 1 so k = 1/18 b) 5/6 c) 10/81
7

A bag contains 3 red counters and 2 blue counters. Ravi takes two counters at random from the bag without replacement. The random variable R is the number of red counters he takes.

  1. Find the probability distribution of R.
  2. Find P(R ≤ 1).
5 marks
Show answer
a) P(R = 0) = 1/10, P(R = 1) = 3/5, P(R = 2) = 3/10 b) 7/10
8

Two fair four-sided dice, each numbered 1 to 4, are rolled. The random variable S is the larger of the two scores (if the scores are equal, S is that score).

  1. Explain why P(S ≤ s) = s216 for s = 1, 2, 3, 4
  2. Hence show that P(S = s) = 2s − 116 for s = 1, 2, 3, 4
  3. Find P(S ≥ 3).
6 marks
Show answer
a) S ≤ s means both dice show s or less, probability (s/4)2 = s2/16 b) P(S = s) = s2/16 - (s - 1)2/16 = (2s - 1)/16 c) 3/4
9

The discrete random variable X has probability function

P(X = x) = kx    for x = 1, 2, 3

P(X = x) = k(7 − x)    for x = 4, 5, 6

where k is a constant.

  1. Find the value of k.
  2. Find P(2 < X ≤ 5).
  3. Find the probability that X is even.
5 marks
Show answer
a) k = 1/12 b) 2/3 c) 1/2
10

In a game, Megan rolls a fair six-sided die until she gets a six, but she stops after 3 rolls even if she has not got a six. The random variable X is the number of times she rolls the die.

  1. Find the probability distribution of X.
  2. Find P(X ≥ 2).
  3. Megan and Tomasz each play the game once, independently. Find the probability that they roll the die the same number of times.
6 marks
Show answer
a) P(X = 1) = 1/6, P(X = 2) = 5/36, P(X = 3) = 25/36 b) 5/6 c) 343/648
11

The random variable X has a discrete uniform distribution over the integers 1, 2, 3, ..., n.

Given that P(X ≤ 6) = 0.4

  1. find the value of n
  2. find P(4 < X < 11)
  3. find the probability that two independent values of X are both greater than 12.
6 marks
Show answer
a) n = 15 b) 2/5 c) 1/25
12

The discrete random variable X has probability function

P(X = x) = kx    for x = 1, 2, 3, 6

  1. Show that k = 12
  2. Find P(X > 1).
  3. Two independent observations X1 and X2 are taken of X. Find P(X1X2 = 6).
  4. A single value g of X is taken and used in the equation t2 + gt + 4 = 0. Find the probability that this equation has real roots.
8 marks
Show answer
a) k(1 + 1/2 + 1/3 + 1/6) = 2k = 1 so k = 1/2 b) 1/2 c) 1/6 d) 1/12

Every question here is original and written by Robinson Tuition. It follows the Edexcel AS specification and suits AQA and OCR AS maths too; this is not an exam-board paper.

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