Independent events across days/trials

CAIE S1 2022 November Q4

4 In a large population, the systolic blood pressure (SBP) of adults is normally distributed with mean 125.4 and standard deviation 18.6.

1. Find the probability that the SBP of a randomly chosen adult is less than 132.
   The SBP of 12-year-old children in the same population is normally distributed with mean 117. Of these children 88\% have SBP more than 108.
2. Find the standard deviation of this distribution.
   Three adults are chosen at random from this population.
3. Find the probability that each of these three adults has SBP within 1.5 standard deviations of the mean.

CAIE S1 2005 June Q6

6 Tyre pressures on a certain type of car independently follow a normal distribution with mean 1.9 bars and standard deviation 0.15 bars.

1. Find the probability that all four tyres on a car of this type have pressures between 1.82 bars and 1.92 bars.
2. Safety regulations state that the pressures must be between \(1.9 - b\) bars and \(1.9 + b\) bars. It is known that \(80 \%\) of tyres are within these safety limits. Find the safety limits.

CAIE S1 2006 June Q3

3 The lengths of fish of a certain type have a normal distribution with mean 38 cm . It is found that \(5 \%\) of the fish are longer than 50 cm .

1. Find the standard deviation.
2. When fish are chosen for sale, those shorter than 30 cm are rejected. Find the proportion of fish rejected.
3. 9 fish are chosen at random. Find the probability that at least one of them is longer than 50 cm .

CAIE S1 2012 June Q7

7 The times taken to play Beethoven's Sixth Symphony can be assumed to have a normal distribution with mean 41.1 minutes and standard deviation 3.4 minutes. Three occasions on which this symphony is played are chosen at random.

1. Find the probability that the symphony takes longer than 42 minutes to play on exactly 1 of these occasions. The times taken to play Beethoven's Fifth Symphony can also be assumed to have a normal distribution. The probability that the time is less than 26.5 minutes is 0.1 , and the probability that the time is more than 34.6 minutes is 0.05 .
2. Find the mean and standard deviation of the times to play this symphony.
3. Assuming that the times to play the two symphonies are independent of each other, find the probability that, when both symphonies are played, both of the times are less than 34.6 minutes.

CAIE S1 2003 November Q7

7 The length of time a person undergoing a routine operation stays in hospital can be modelled by a normal distribution with mean 7.8 days and standard deviation 2.8 days.

1. Calculate the proportion of people who spend between 7.8 days and 11.0 days in hospital.
2. Calculate the probability that, of 3 people selected at random, exactly 2 spend longer than 11.0 days in hospital.
3. A health worker plotted a box-and-whisker plot of the times that 100 patients, chosen randomly, stayed in hospital. The result is shown below. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{26776153-9477-4155-b5e4-f35e6d33a5ff-3_447_917_767_657} \captionsetup{labelformat=empty} \caption{Days}
   \end{figure} State with a reason whether or not this agrees with the model used in parts (i) and (ii).

CAIE S1 2004 November Q5

5 The length of Paulo's lunch break follows a normal distribution with mean \(\mu\) minutes and standard deviation 5 minutes. On one day in four, on average, his lunch break lasts for more than 52 minutes.

1. Find the value of \(\mu\).
2. Find the probability that Paulo's lunch break lasts for between 40 and 46 minutes on every one of the next four days.

CAIE S1 2009 November Q7

7 The weights, \(X\) grams, of bars of soap are normally distributed with mean 125 grams and standard deviation 4.2 grams.

1. Find the probability that a randomly chosen bar of soap weighs more than 128 grams.
2. Find the value of \(k\) such that \(\mathrm { P } ( k < X < 128 ) = 0.7465\).
3. Five bars of soap are chosen at random. Find the probability that more than two of the bars each weigh more than 128 grams.

CAIE S1 2012 November Q2

2 The random variable \(X\) is the daily profit, in thousands of dollars, made by a company. \(X\) is normally distributed with mean 6.4 and standard deviation 5.2.

1. Find the probability that, on a randomly chosen day, the company makes a profit between \(
   ) 10000\( and \)\\( 12000\).
2. Find the probability that the company makes a loss on exactly 1 of the next 4 consecutive days.

CAIE S1 2013 November Q5

5 Lengths of a certain type of carrot have a normal distribution with mean 14.2 cm and standard deviation 3.6 cm .

1. \(8 \%\) of carrots are shorter than \(c \mathrm {~cm}\). Find the value of \(c\).
2. Rebekah picks 7 carrots at random. Find the probability that at least 2 of them have lengths between 15 and 16 cm .

CAIE S1 2013 November Q3

3 The amount of fibre in a packet of a certain brand of cereal is normally distributed with mean 160 grams. 19\% of packets of cereal contain more than 190 grams of fibre.

1. Find the standard deviation of the amount of fibre in a packet.
2. Kate buys 12 packets of cereal. Find the probability that at least 1 of the packets contains more than 190 grams of fibre.

CAIE S1 2014 November Q3

3

1. Four fair six-sided dice, each with faces marked \(1,2,3,4,5,6\), are thrown. Find the probability that the numbers shown on the four dice add up to 5 .
2. Four fair six-sided dice, each with faces marked \(1,2,3,4,5,6\), are thrown on 7 occasions. Find the probability that the numbers shown on the four dice add up to 5 on exactly 1 or 2 of the 7 occasions. Sharik attempts a multiple choice revision question on-line. There are 3 suggested answers, one of which is correct. When Sharik chooses an answer the computer indicates whether the answer is right or wrong. Sharik first chooses one of the three suggested answers at random. If this answer is wrong he has a second try, choosing an answer at random from the remaining 2 . If this answer is also wrong Sharik then chooses the remaining answer, which must be correct.

CAIE S1 2016 November Q4

4 The time taken to cook an egg by people living in a certain town has a normal distribution with mean 4.2 minutes and standard deviation 0.6 minutes.

1. Find the probability that a person chosen at random takes between 3.5 and 4.5 minutes to cook an egg.
   \(12 \%\) of people take more than \(t\) minutes to cook an egg.
2. Find the value of \(t\).
3. A random sample of \(n\) people is taken. Find the smallest possible value of \(n\) if the probability that none of these people takes more than \(t\) minutes to cook an egg is less than 0.003 .

CAIE S1 2017 November Q7

7 Josie aims to catch a bus which departs at a fixed time every day. Josie arrives at the bus stop \(T\) minutes before the bus departs, where \(T \sim \mathrm {~N} \left( 5.3,2.1 ^ { 2 } \right)\).

1. Find the probability that Josie has to wait longer than 6 minutes at the bus stop.
   On \(5 \%\) of days Josie has to wait longer than \(x\) minutes at the bus stop.
2. Find the value of \(x\).
3. Find the probability that Josie waits longer than \(x\) minutes on fewer than 3 days in 10 days.
4. Find the probability that Josie misses the bus.

Edexcel S1 2024 January Q5

1. The distance an athlete can throw a discus is normally distributed with mean 40 m and standard deviation 4 m
   1. Using standardisation, show that the probability that this athlete throws the discus less than 38.8 m is 0.3821
   This athlete enters a discus competition.
   To qualify for the final, they have 3 attempts to throw the discus a distance of more than 38.8 m
   Once they qualify, they do not use any of their remaining attempts.
   Given that they qualified for the final and that throws are independent,
2. find the probability that this athlete qualified for the final on their second throw with a distance of more than 44 m

OCR MEI S2 2013 January Q3

3 The amount of data, \(X\) megabytes, arriving at an internet server per second during the afternoon is modelled by the Normal distribution with mean 435 and standard deviation 30.

1. Find
   (A) \(\mathrm { P } ( X < 450 )\),
   (B) \(\mathrm { P } ( 400 < X < 450 )\).
2. Find the probability that, during 5 randomly selected seconds, the amounts of data arriving are all between 400 and 450 megabytes. The amount of data, \(Y\) megabytes, arriving at the server during the evening is modelled by the Normal distribution with mean \(\mu\) and standard deviation \(\sigma\).
3. Given that \(\mathrm { P } ( Y < 350 ) = 0.2\) and \(\mathrm { P } ( Y > 390 ) = 0.1\), find the values of \(\mu\) and \(\sigma\).
4. Find values of \(a\) and \(b\) for which \(\mathrm { P } ( a < Y < b ) = 0.95\).

OCR MEI S2 2010 June Q3

3 In a men's cycling time trial, the times are modelled by the random variable \(X\) minutes which is Normally distributed with mean 63 and standard deviation 5.2.

1. Find $$\begin{aligned} & \text { (A) } \mathrm { P } ( X < 65 ) \text {, }
   & \text { (B) } \mathrm { P } ( 60 < X < 65 ) \text {. } \end{aligned}$$
2. Find the probability that 5 riders selected at random all record times between 60 and 65 minutes.
3. A competitor aims to be in the fastest \(5 \%\) of entrants (i.e. those with the lowest times). Find the maximum time that he can take. It is suggested that holding the time trial on a new course may result in lower times. To investigate this, a random sample of 15 competitors is selected. These 15 competitors do the time trial on the new course. The mean time taken by these riders is 61.7 minutes. You may assume that times are Normally distributed and the standard deviation is still 5.2 minutes. A hypothesis test is carried out to investigate whether times on the new course are lower.
4. Write down suitable null and alternative hypotheses for the test. Carry out the test at the 5\% significance level.

OCR MEI S2 2011 June Q3

3 The weights of Braeburn apples on display in a supermarket, measured in grams, are Normally distributed with mean 210.5 and standard deviation 15.2.

1. Find the probability that a randomly selected apple weighs at least 220 grams.
2. These apples are sold in packs of 3. You may assume that the weights of apples in each pack are independent. Find the probability that all 3 of the apples in a randomly selected pack weigh at least 220 grams.
3. 100 packs are selected at random.
   (A) State the exact distribution of the number of these 100 packs in which all 3 apples weigh at least 220 grams.
   (B) Use a suitable approximating distribution to find the probability that in at most one of these packs all 3 apples weigh at least 220 grams.
   (C) Explain why this approximating distribution is suitable.
4. The supermarket also sells Cox's Orange Pippin apples. The weights of these apples, measured in grams, are Normally distributed with mean 185 and standard deviation \(\sigma\).
   (A) Given that \(10 \%\) of randomly selected Cox's Orange Pippin apples weigh less than 170 grams, calculate the value of \(\sigma\).
   (B) Sketch the distributions of the weights of both types of apple on a single diagram.

OCR MEI S2 2012 June Q3

3 At a vineyard, the process used to fill bottles with wine is subject to variation. The contents of bottles are independently Normally distributed with mean \(\mu = 751.4 \mathrm { ml }\) and standard deviation \(\sigma = 2.5 \mathrm { ml }\).

1. Find the probability that a randomly selected bottle contains at least 750 ml .
2. A case of wine consists of 6 bottles. Find the probability that all 6 bottles in a case contain at least 750 ml .
3. Find the probability that, in a random sample of 25 cases, there are at least 2 cases in which all 6 bottles contain at least 750 ml . It is decided to increase the proportion of bottles which contain at least 750 ml to \(98 \%\).
4. This can be done by changing the value of \(\mu\), but retaining the original value of \(\sigma\). Find the required value of \(\mu\).
5. An alternative is to change the value of \(\sigma\), but retain the original value of \(\mu\). Find the required value of \(\sigma\).
6. Comment briefly on which method might be easier to implement and which might be preferable to the vineyard owners.

OCR H240/02 Q7

7

1. The heights of English men aged 25 to 34 are normally distributed with mean 178 cm and standard deviation 8 cm .
   Three English men aged 25 to 34 are chosen at random. Find the probability that all three men have a height less than 194 cm .
2. The diagram shows the distribution of heights of Scottish women aged 25 to 34.
   \includegraphics[max width=\textwidth, alt={}, center]{f2f45d6c-cfdc-455b-ab08-597b06a69f36-08_585_1477_909_342} The distribution is approximately normal. Use the diagram in the Printed Answer Booklet to estimate the standard deviation of these heights, explaining your method.

Edexcel S1 2012 January Q7

1. A manufacturer fills jars with coffee. The weight of coffee, \(W\) grams, in a jar can be modelled by a normal distribution with mean 232 grams and standard deviation 5 grams.
   1. Find \(\mathrm { P } ( W < 224 )\).
   2. Find the value of \(w\) such that \(\mathrm { P } ( 232 < W < w ) = 0.20\)
   Two jars of coffee are selected at random.
2. Find the probability that only one of the jars contains between 232 grams and \(w\) grams of coffee.

Edexcel S1 2008 June Q7

7. A packing plant fills bags with cement. The weight \(X \mathrm {~kg}\) of a bag of cement can be modelled by a normal distribution with mean 50 kg and standard deviation 2 kg .

1. Find \(\mathrm { P } ( X > 53 )\).
2. Find the weight that is exceeded by \(99 \%\) of the bags. Three bags are selected at random.
3. Find the probability that two weigh more than 53 kg and one weighs less than 53 kg .

Edexcel S1 2002 November Q4

4. Strips of metal are cut to length \(L \mathrm {~cm}\), where \(L \sim \mathrm {~N} \left( \mu , 0.5 ^ { 2 } \right)\).

1. Given that \(2.5 \%\) of the cut lengths exceed 50.98 cm , show that \(\mu = 50\).
2. Find \(\mathrm { P } ( 49.25 < L < 50.75 )\). Those strips with length either less than 49.25 cm or greater than 50.75 cm cannot be used.
   Two strips of metal are selected at random.
3. Find the probability that both strips cannot be used.

AQA S1 2010 June Q3

3 Each day, Margot completes the crossword in her local morning newspaper. Her completion times, \(X\) minutes, can be modelled by a normal random variable with a mean of 65 and a standard deviation of 20 .

1. Determine:
   1. \(\mathrm { P } ( X < 90 )\);
   2. \(\mathrm { P } ( X > 60 )\).
2. Given that Margot's completion times are independent from day to day, determine the probability that, during a particular period of 6 days:
   1. she completes one of the six crosswords in exactly 60 minutes;
   2. she completes each crossword in less than 60 minutes;
   3. her mean completion time is less than 60 minutes.
      
      \includegraphics[max width=\textwidth, alt={}]{c4844a30-6a86-49e3-b6aa-8e213dfc8ca1-07_2484_1709_223_153}

AQA S1 2011 June Q2

2 The diameter, \(D\) millimetres, of an American pool ball may be modelled by a normal random variable with mean 57.15 and standard deviation 0.04 .

1. Determine:
   1. \(\mathrm { P } ( D < 57.2 )\);
   2. \(\mathrm { P } ( 57.1 < D < 57.2 )\).
2. A box contains 16 of these pool balls. Given that the balls may be regarded as a random sample, determine the probability that:
   1. all 16 balls have diameters less than 57.2 mm ;
   2. the mean diameter of the 16 balls is greater than 57.16 mm .

Edexcel S1 Q5

5. The time taken in minutes, \(T\), for a mechanic to service a bicycle follows a normal distribution with a mean of 25 minutes and a variance of 16 minutes \(^ { 2 }\). Find

1. \(\mathrm { P } ( T < 28 )\),
2. \(\quad \mathrm { P } ( | T - 25 | < 5 )\). One afternoon the mechanic has 3 bicycles to service.
3. Find the probability that he will take less than 23 minutes on each of the three bicycles.
   (4 marks)