The normal distribution: past paper questions

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• Question 6May/June 2026, paper 51Structured12 marks

  The heights of adult Indian elephants are normally distributed with mean $2.74\text{ m}$ and standard deviation $0.25\text{ m}$. A random sample of 250 of these elephants is chosen.

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• 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.

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• Question 5Oct/Nov 2024, paper 51Structured9 marks

  The weights of the green apples sold by a shop are normally distributed with mean 90 grams and standard deviation 8 grams. Find the probability that a randomly chosen green apple weighs between 83 grams and 95 grams.

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• Question 3Oct/Nov 2023, paper 51Structured11 marks

  A farmer sells eggs. The weights, in grams, of the eggs can be modelled by a normal distribution with mean 80.5 and standard deviation 6.6. Eggs are classified as small, medium or large according to their weight.

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

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

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• Question 6Oct/Nov 2025, paper 51Structured9 marks

  A factory produces chocolate bars. The weights of the bars are normally distributed with mean $155\text{ g}$ and standard deviation $6\text{ g}$. A random sample of 350 of these bars is chosen.

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• 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.

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• 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.

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• Question 7May/June 2026, paper 52Structured9 marks

  The lengths of the leaves of a certain type of tree are normally distributed with mean $\mu \mathrm{c}\mathrm{m}$ and standard deviation $\sigma \mathrm{c}\mathrm{m}$.

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• Question 3Oct/Nov 2025, paper 52Structured7 marks

  The heights of the 124 Senior members of the Giraffes basketball club are normally distributed with mean $187.4\text{ cm}$ and standard deviation $6.4\text{ cm}$. How many members of the club would you expect to have heights within $5\text{ cm}$ of the mean?

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• Question 4Oct/Nov 2024, paper 52Structured7 marks

  The heights, in metres, of white pine trees are normally distributed with mean 19.8 and standard deviation 2.4 . In a certain forest there are 450 white pine trees. How many of these trees would you expect to have height less than 18.2 metres?

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• Question 5Oct/Nov 2023, paper 52Structured8 marks

  The heights of the members of a club are normally distributed with mean 166 cm and standard deviation 10 cm. Find the probability that a randomly chosen member of the club has height less than 170 cm.

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• Question 5May/June 2026, paper 53Structured11 marks

  The lengths of Indian cobra snakes are normally distributed with mean $125\text{ cm}$ and standard deviation $8\text{ cm}$. Find the probability that a randomly chosen Indian cobra has a length between $105\text{ cm}$ and $135\text{ cm}$.

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• 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.

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• 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.

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• Question 2Oct/Nov 2023, paper 53Structured6 marks

  The weights of large bags of pasta produced by a company are normally distributed with mean $1.5\text{ kg}$ and standard deviation $0.05\text{ kg}$. Find the probability that a randomly chosen large bag of pasta weighs between $1.42\text{ kg}$ and $1.52\text{ kg}$.

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• Question 2May/June 2026, paper 62Structured5 marks

  Each week, Birgitte takes three train journeys. The times, in minutes, for these journeys have the independent distributions $A\sim \mathrm{N}(28,15)$, $B\sim \mathrm{N}(23,8)$ and $C\sim \mathrm{N}(99,24)$. For a randomly chosen week, find $\mathrm{P}(C>2(A+B))$.

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• Question 7Oct/Nov 2025, paper 53Structured10 marks

  The times taken by Obi to walk to work each morning are normally distributed with mean 14.8 minutes and standard deviation 1.5 minutes. Find the probability that, on a randomly chosen day, Obi takes more than 15.6 minutes to walk to work.

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• 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.

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• 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.

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• Question 1May/June 2025, paper 51Structured3 marks

  The masses of the bags of rice made by a company are normally distributed with mean $\mu \text{ kg}$ and standard deviation $0.14\text{ kg}$. The probability that the mass of a randomly chosen bag of this rice is less than $1.48\text{ kg}$ is $0.22$. Find the value of $\mu$.

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• Question 3Oct/Nov 2024, paper 53Structured7 marks

  In Molimba, the heights, in cm, of adult males are normally distributed with mean $176\text{ cm}$ and standard deviation $4.8\text{ cm}$. Find the probability that a randomly chosen adult male in Molimba has a height greater than $170\text{ cm}$.

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• Question 6Oct/Nov 2023, paper 62Structured7 marks

  A factory makes loaves of bread in batches. One batch of loaves contains $X$ kilograms of dried yeast and $Y$ kilograms of flour, where $X$ and $Y$ have the independent distributions $\mathrm{N}(0.7,{0.02}^2)$ and $\mathrm{N}(100.0,{3.0}^2)$ respectively.

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• Question 4May/June 2025, paper 51Structured12 marks

  Every Saturday, a particular community holds a ‘Puzzle’ event to raise money for a new Leisure Centre. Competitors attempt to solve a puzzle as quickly as possible.

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• Question 2May/June 2024, paper 51Structured7 marks

  The lengths of the tails of adult raccoons of a certain species are normally distributed with mean $28\text{ cm}$ and standard deviation $3.3\text{ cm}$.

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• Question 4May/June 2023, paper 51Structured9 marks

  A mathematical puzzle is given to a large number of students. The times taken to complete the puzzle are normally distributed with mean 14.6 minutes and standard deviation 5.2 minutes.

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• Question 2May/June 2025, paper 52Structured5 marks

  In Millford, 70% of the residents own a bicycle. A random sample of 160 residents is selected. Use a suitable approximation to find the probability that more than 120 of these residents own a bicycle.

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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 6May/June 2023, paper 51Structured11 marks

  Eli has four fair 4-sided dice with sides labelled 1, 2, 3, 4. He throws all four dice at the same time. The random variable $X$ denotes the number of 2s obtained.

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• Question 7May/June 2025, paper 52Structured12 marks

  Kestrels are birds whose adult wingspans are normally distributed with mean $74.8\text{ cm}$ and standard deviation $3.2\text{ cm}$. A random sample of 120 adult kestrels is selected.

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