Find p then binomial probability

Given a normal distribution with unknown parameter(s), find the parameter(s) using a given probability or percentile, then use a probability from this distribution as p in a binomial calculation to find probabilities of events in repeated trials.

14 questions · Standard +0.3

2.04e Normal distribution: as model N(mu, sigma^2)2.04f Find normal probabilities: Z transformation
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CAIE S1 2023 June Q6
10 marks Standard +0.3
6 The mass of grapes sold per day by a large shop can be modelled by a normal distribution with mean 28 kg . On \(10 \%\) of days less than 16 kg of grapes are sold.
  1. Find the standard deviation of the mass of grapes sold per day.
    The mass of grapes sold on any day is independent of the mass sold on any other day.
  2. 12 days are chosen at random. Find the probability that less than 16 kg of grapes are sold on more than 2 of these 12 days.
  3. In a random sample of 365 days, on how many days would you expect the mass of grapes sold to be within 1.3 standard deviations of the mean?
CAIE S1 2021 November Q7
11 marks Moderate -0.3
7 The times, in minutes, that Karli spends each day on social media are normally distributed with mean 125 and standard deviation 24.
    1. On how many days of the year ( 365 days) would you expect Karli to spend more than 142 minutes on social media?
    2. Find the probability that Karli spends more than 142 minutes on social media on fewer than 2 of 10 randomly chosen days.
  2. On \(90 \%\) of days, Karli spends more than \(t\) minutes on social media. Find the value of \(t\).
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
CAIE S1 2006 June Q3
8 marks Standard +0.3
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 2011 June Q5
11 marks Standard +0.8
5 The random variable \(X\) is normally distributed with mean \(\mu\) and standard deviation \(\frac { 1 } { 4 } \mu\). It is given that \(\mathrm { P } ( X > 20 ) = 0.04\).
  1. Find \(\mu\).
  2. Find \(\mathrm { P } ( 10 < X < 20 )\).
  3. 250 independent observations of \(X\) are taken. Find the probability that at least 235 of them are less than 20.
CAIE S1 2004 November Q5
7 marks Standard +0.8
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 2013 November Q3
5 marks Standard +0.3
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 2010 November Q7
14 marks Standard +0.3
7 The times spent by people visiting a certain dentist are independent and normally distributed with a mean of 8.2 minutes. \(79 \%\) of people who visit this dentist have visits lasting less than 10 minutes.
  1. Find the standard deviation of the times spent by people visiting this dentist.
  2. Find the probability that the time spent visiting this dentist by a randomly chosen person deviates from the mean by more than 1 minute.
  3. Find the probability that, of 6 randomly chosen people, more than 2 have visits lasting longer than 10 minutes.
  4. Find the probability that, of 35 randomly chosen people, fewer than 16 have visits lasting less than 8.2 minutes.
CAIE S1 2011 November Q5
9 marks Standard +0.3
5 The weights of letters posted by a certain business are normally distributed with mean 20 g . It is found that the weights of \(94 \%\) of the letters are within 12 g of the mean.
  1. Find the standard deviation of the weights of the letters.
  2. Find the probability that a randomly chosen letter weighs more than 13 g .
  3. Find the probability that at least 2 of a random sample of 7 letters have weights which are more than 12 g above the mean.
CAIE S1 2012 November Q3
6 marks Standard +0.3
3 Lengths of rolls of parcel tape have a normal distribution with mean 75 m , and 15\% of the rolls have lengths less than 73 m .
  1. Find the standard deviation of the lengths. Alison buys 8 rolls of parcel tape.
  2. Find the probability that fewer than 3 of these rolls have lengths more than 77 m .
Edexcel S2 2024 January Q2
8 marks Standard +0.3
  1. The length of pregnancy for a randomly selected pregnant sheep is \(D\) days where
$$D \sim \mathrm {~N} \left( 112.4 , \sigma ^ { 2 } \right)$$ Given that 5\% of pregnant sheep have a length of pregnancy of less than 108 days,
  1. find the value of \(\sigma\) Qiang selects 25 pregnant sheep at random from a large flock.
  2. Find the probability that more than 3 of these pregnant sheep have a length of pregnancy of less than 108 days. Charlie takes 200 random samples of 25 pregnant sheep.
  3. Use a Poisson approximation to estimate the probability that at least 2 of the samples have more than 3 pregnant sheep with a length of pregnancy of less than 108 days.
Edexcel S1 Q7
17 marks Standard +0.3
7. The times taken by a large number of people to read a certain book can be modelled by a normal distribution with mean \(5 \cdot 2\) hours. It is found that \(62 \cdot 5 \%\) of the people took more than \(4 \cdot 5\) hours to read the book.
  1. Show that the standard deviation of the times is approximately \(2 \cdot 2\) hours.
  2. Calculate the percentage of the people who took between 4 and 7 hours to read the book.
  3. Calculate the probability that two of the people chosen at random both took less than 5 hours to read the book, stating any assumption that you make.
  4. If a number of extra people were taken into account, all of whom took exactly \(5 \cdot 2\) hours to read the book, state with reasons what would happen to (i) the mean, (ii) the variance and explain briefly why the distribution would no longer be normal.
CAIE S1 2002 June Q6
8 marks Standard +0.3
  1. In a normal distribution with mean \(\mu\) and standard deviation \(\sigma\), \(\text{P}(X > 3.6) = 0.5\) and \(\text{P}(X > 2.8) = 0.6554\). Write down the value of \(\mu\), and calculate the value of \(\sigma\). [4]
  2. If four observations are taken at random from this distribution, find the probability that at least two observations are greater than 2.8. [4]
AQA Paper 3 2019 June Q17
12 marks Standard +0.3
Elizabeth's Bakery makes brownies. It is known that the mass, \(X\) grams, of a brownie may be modelled by a normal distribution. 10\% of the brownies have a mass less than 30 grams. 80\% of the brownies have a mass greater than 32.5 grams.
  1. Find the mean and standard deviation of \(X\). [7 marks]
    1. Find P\((X \neq 35)\) [1 mark]
    2. Find P\((X < 35)\) [2 marks]
  3. Brownies are baked in batches of 13. Calculate the probability that, in a batch of brownies, no more than 3 brownies are less than 35 grams. You may assume that the masses of brownies are independent of each other. [2 marks]
SPS SPS FM Statistics 2021 June Q4
9 marks Standard +0.3
The weights of sacks of potatoes are normally distributed. It is known that one in five sacks weigh more than 6kg and three in five sacks weigh more than 5.5kg.
  1. Find the mean and standard deviation of the weights of potato sacks. [5]
  2. The sacks are put into crates, with twelve sacks going into each crate. What is the probability that a given crate contains two or more sacks that weigh more than 6kg? You must explain your reasoning clearly in this question. [4]