Discrete random variables: past paper questions

Where the questions come from

• Question 1May/June 2026, paper 51Structured6 marks

  A fair red 4-sided spinner has sides numbered 1, 2, 3, 3. A fair blue 4-sided spinner has sides numbered $-2$, $-1$, $-1$, 1. The two spinners are spun and the numbers on the sides on which they land are noted.

  Try this question
• Question 1Oct/Nov 2025, paper 51Structured6 marks

  The random variable $X$ takes the value $x$ with probability $k{x}^2$, where $k$ is a constant and $x$ takes the values $-2,1,2,3$ only. Draw up the probability distribution table for $X$, giving the probabilities as numerical fractions.

  Try this question
• Question 1Oct/Nov 2024, paper 51Structured4 marks

  Nicola throws an ordinary fair six-sided dice. The random variable $X$ is the number of throws that she takes to obtain a 6. Find $\mathrm{P}(X<8)$.

  Try this question
• Question 2Oct/Nov 2023, paper 51Structured7 marks

  Hazeem repeatedly throws two ordinary fair 6-sided dice at the same time. On each occasion, the score is the sum of the two numbers that she obtains. Find the probability that it takes exactly 5 throws of the two dice for Hazeem to obtain a score of 8 or more.

  Try this question
• Question 4May/June 2026, paper 51Structured8 marks

  Jong has a bag containing 5 red marbles and 7 blue marbles. He chooses marbles from the bag one at a time at random. Each time he replaces the marble in the bag.

  Try this question
• Question 2Oct/Nov 2025, paper 51Structured4 marks

  A fair six-sided dice with faces labelled 1, 2, 3, 4, 5, 6 is thrown repeatedly until a 6 is obtained. Find the probability that a 6 is obtained for the first time on the 8th throw.

  Try this question
• Question 2Oct/Nov 2024, paper 51Structured6 marks

  The random variable $X$ takes the values $-2,-1,0,2,3$. It is given that $\mathrm{P}(X=x)=k(x^2+2)$, where $k$ is a positive constant. Draw up the probability distribution table for $X$, giving the probabilities as numerical fractions.

  Try this question
• Question 5Oct/Nov 2023, paper 51Structured9 marks

  A red spinner has four sides labelled 1, 2, 3, 4. When the spinner is spun, the score is the number on the side on which it lands. The random variable $X$ denotes this score.

  Try this question
• Question 2May/June 2026, paper 52Structured8 marks

  Ling has three coins. Each coin is biased so that the probability of obtaining a head is $\frac{1}{3}$. Ling throws all three coins at the same time.

  Try this question
• Question 5Oct/Nov 2025, paper 51Structured8 marks

  On any given day, Cooper either wears a blue jumper or he wears a green jumper or he does not wear a jumper. The probability that he wears a blue jumper is 0.6 and the probability that he wears a green jumper is 0.3.

  Try this question
• Question 6Oct/Nov 2024, paper 51Structured8 marks

  The heights of the female students at Breven college are normally distributed: • 90% of the female students have heights less than 182.7 cm. • 40% of the female students have heights less than 162.5 cm.

  Try this question
• Question 1Oct/Nov 2023, paper 52Structured5 marks

  A competitor in a throwing event has three attempts to throw a ball as far as possible. The random variable $X$ denotes the number of throws that exceed 30 metres. The probability distribution table for $X$ is shown below.

  Try this question
• Question 4May/June 2026, paper 52Structured8 marks

  Every day Haz attempts to complete a particular type of puzzle in less than 10 minutes.

  Try this question
• Question 1Oct/Nov 2025, paper 52Structured3 marks

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

  Try this question
• Question 1Oct/Nov 2024, paper 52Structured6 marks

  At a college, the students choose exactly one of tennis, hockey or netball to play. The table shows the numbers of students in Year 1 and Year 2 at the college playing each of these sports.

  Try this question
• Question 2Oct/Nov 2023, paper 52Structured5 marks

  George has a fair 5-sided spinner with sides labelled 1, 2, 3, 4, 5. He spins the spinner and notes the number on the side on which the spinner lands. Find the probability that it takes fewer than 7 spins for George to obtain a 5.

  Try this question
• Question 1May/June 2026, paper 53Structured6 marks

  A fair 8-sided dice with faces labelled 1, 2, 3, 4, 5, 6, 7, 8 is thrown repeatedly. Find the probability that a 5 will be obtained for the first time on the 7th throw.

  Try this question
• Question 2Oct/Nov 2025, paper 52Structured7 marks

  Kayla has a bag containing 3 red marbles, 1 blue marble and 2 green marbles. She selects one marble from the bag at random and does not replace it in the bag. She repeats this process until she obtains a green marble.

  Try this question
• Question 3Oct/Nov 2024, paper 52Structured6 marks

  A fair coin and an ordinary fair six-sided dice are thrown at the same time. The random variable $X$ is defined as follows. • If the coin shows a tail, $X$ is twice the score on the dice.

  Try this question
• Question 3Oct/Nov 2023, paper 52Structured5 marks

  A factory produces a certain type of electrical component. It is known that 15% of the components produced are faulty. A random sample of 200 components is chosen.

  Try this question
• Question 2May/June 2026, paper 53Structured6 marks

  The random variable $X$ takes the values $-2,0,1,3$. It is given that $\mathrm{P}(X=x)=k(x+1{)}^2$, where $k$ is a constant. Draw up the probability distribution table for $X$, giving the probabilities as numerical fractions not involving $k$.

  Try this question
• Question 6Oct/Nov 2025, paper 52Structured9 marks

  For a randomly chosen person, their next birthday is equally likely to occur on any day of the week, independently of any other person's birthday. Find the probability that, out of 10 randomly chosen people, none of them will have their next birthday on a Saturday or Sunday.

  Try this question
• Question 7Oct/Nov 2024, paper 52Structured11 marks

  In a game, players attempt to score a goal by kicking a ball into a net. The probability that Leno scores a goal is 0.4 on any attempt, independently of all other attempts. The random variable $X$ denotes the number of attempts that it takes Leno to score a goal.

  Try this question
• Question 1Oct/Nov 2023, paper 53Structured6 marks

  Becky sometimes works in an office and sometimes works at home. The random variable $X$ denotes the number of days that she works at home in any given week. It is given that $$\mathrm{P}(X=x)=kx(x+1),$$ where $k$ is a constant and $x=1,2,3\text{ or }4$ only.

  Try this question
• Question 4May/June 2026, paper 61Structured11 marks

  The score on one spin of a 5-sided spinner is denoted by the random variable $X$ with probability distribution as shown in the table.

  Try this question
• Question 1Oct/Nov 2025, paper 53Structured4 marks

  There are a large number of students at Greenfield college. Each student travels to college by car, by bus or on foot, independently of any other student. The probability that any student travels by car is 0.4.

  Try this question
• Question 1Oct/Nov 2024, paper 53Structured7 marks

  30% of the residents of Wimfield own an electric car. Three residents are chosen at random. Find the probability that either all three own an electric car or none of them owns an electric car.

  Try this question
• Question 5Oct/Nov 2023, paper 53Structured11 marks

  The probability that a driver passes an advanced driving test is 0.3 on any given attempt. Dipak keeps taking the test until he passes. The random variable $X$ denotes the number of attempts required for Dipak to pass the test.

  Try this question
• Question 4Oct/Nov 2025, paper 53Structured8 marks

  A fair red spinner has 4 sides, numbered 1, 2, 3, 4. A fair blue spinner has 4 sides, numbered 0, 1, 2, 3. When a spinner is spun, the score is the number on the side on which it lands.

  Try this question
• Question 2Oct/Nov 2024, paper 53Structured4 marks

  A red fair six-sided dice has faces labelled 1, 1, 1, 2, 2, 2. A blue fair six-sided dice has faces labelled 1, 1, 2, 2, 3, 3. Both dice are thrown.

  Try this question