Probability: past paper questions

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• Question 2May/June 2026, paper 51Structured5 marks

  At a certain college, each of the 150 students studies exactly one of Art, Business or Drama. The numbers of male and female students studying each of these subjects is given in the table.

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• 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.

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• 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 $\mathrm{P}(X<8)$.

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• 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.

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• 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.

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• 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.

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• Question 4Oct/Nov 2024, paper 51Structured6 marks

  Rahul has two bags, $X$ and $Y$. Bag $X$ contains 4 red marbles and 2 blue marbles. Bag $Y$ contains 3 red marbles and 4 blue marbles.

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• 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.

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• Question 5May/June 2026, paper 51Structured10 marks

  Three letters are chosen at random from the 9 letters in the word RETRIEVES. Find the probability that at least one of the letters is an E.

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• Question 4Oct/Nov 2025, paper 51Structured5 marks

  Bag $A$ contains 8 red marbles and 3 blue marbles. Bag $B$ contains 4 red marbles and 1 blue marble. A marble is chosen at random from bag $A$.

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• 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.

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• Question 6Oct/Nov 2023, paper 52Structured9 marks

  Freddie has two bags of marbles. Bag $X$ contains 7 red marbles and 3 blue marbles. Bag $Y$ contains 4 red marbles and 1 blue marble.

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• Question 1May/June 2026, paper 52Structured6 marks

  A club has 15 members made up of 6 men and 9 women. Every Friday a team of 5 members is selected at random to represent the club in a competition. Find the probability that next Friday the team will consist of more men than women.

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• Question 7Oct/Nov 2025, paper 51Structured10 marks

  How many different arrangements are there of the 10 letters in the word SEYCHELLES?

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• Question 5Oct/Nov 2024, paper 53Structured10 marks

  A factory produces chocolates. 30% of the chocolates are wrapped in gold foil, 25% are wrapped in red foil and the remainder are unwrapped. Indigo chooses 8 chocolates at random from the production line.

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• Question 7Oct/Nov 2023, paper 52Structured10 marks

  Find the number of different arrangements of the 9 letters in the word ANDROMEDA in which no consonant is next to another consonant.

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• 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.

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• 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.

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• Question 6Oct/Nov 2024, paper 53Structured11 marks

  Find the number of different arrangements of the 9 letters in the word HAPPINESS.

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• Question 3Oct/Nov 2023, paper 53Structured7 marks

  Tim has two bags of marbles, $A$ and $B$. Bag $A$ contains 8 white, 4 red and 3 yellow marbles. Bag $B$ contains 6 white, 7 red and 2 yellow marbles.

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• Question 5May/June 2026, paper 52Structured7 marks

  A box contains 3 red pens, 5 blue pens and 2 green pens. Lui chooses one pen at random from the box and keeps it. A green pen is added to the box.

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• Question 4Oct/Nov 2025, paper 52Structured7 marks

  Gio has a pack of 18 cards. Ivy has a pack of $x$ cards. Each card has a picture of a bus or a car or a train.

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• Question 4May/June 2024, paper 51Structured6 marks

  A game for two players is played using a fair 4-sided dice with sides numbered 1, 2, 3 and 4. One turn consists of throwing the dice repeatedly up to a maximum of three times. When a 4 is obtained, no further throws are made during that turn.

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• Question 6Oct/Nov 2023, paper 53Structured10 marks

  Jai and his wife Kaz are having a party. Jai has invited five friends and each friend will bring his wife. At the beginning of the party, the 12 people will stand in a line for a photograph.

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• 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.

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• Question 7Oct/Nov 2025, paper 52Structured10 marks

  Find the number of different arrangements of the 10 letters in the word ZOOLOGICAL in which the three Os are together and the two Ls are not next to each other.

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• Question 5May/June 2024, paper 51Structured10 marks

  In a certain area in the Arctic the probability that it snows on any given day is 0.7, independent of all other days. Find the probability that in a week (7 days) it snows on at least five days.

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• Question 3May/June 2023, paper 51Structured6 marks

  Find the number of different arrangements of the 8 letters in the word COCOONED.

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• Question 4May/June 2026, paper 53Structured9 marks

  Ebere has a coin which is biased so that the probability of obtaining a head is $\frac{1}{4}$. She also has two bags containing coloured discs. Bag $A$ contains 5 red discs and 3 blue discs.

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• 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.

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