Conditional probability with normal

Find P(A|B) where both events involve a normally distributed variable, using the definition of conditional probability.

11 questions · Standard +0.4

2.04e Normal distribution: as model N(mu, sigma^2)2.04f Find normal probabilities: Z transformation
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Edexcel S1 2016 January Q5
14 marks Standard +0.3
5. Rosie keeps bees. The amount of honey, in kg, produced by a hive of Rosie's bees in a season, is modelled by a normal distribution with a mean of 22 kg and a standard deviation of 10 kg .
  1. Find the probability that a hive of Rosie's bees produces less than 18 kg of honey in a season. The local bee keepers' club awards a certificate to every hive that produces more than 39 kg of honey in a season, and a medal to every hive that produces more than 50 kg in a season. Given that one of Rosie's bee hives is awarded a certificate
  2. find the probability that this hive is also awarded a medal.
    (5) Sam also keeps bees. The amount of honey, in kg, produced by a hive of Sam's bees in a season, is modelled by a normal distribution with mean \(\mu \mathrm { kg }\) and standard deviation \(\sigma \mathrm { kg }\). The probability that a hive of Sam's bees produces less than 28 kg of honey in a season is 0.8413 Only 20\% of Sam's bee hives produce less than 18 kg of honey in a season.
  3. Find the value of \(\mu\) and the value of \(\sigma\). Give your answers to 2 decimal places.
    (6)
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Edexcel S1 2019 January Q3
10 marks Standard +0.3
  1. The weights of women boxers in a tournament are normally distributed with mean 64 kg and standard deviation 8 kg .
    1. Find the probability that a randomly chosen woman boxer in the tournament weighs less than 51 kg .
    In the tournament, women boxers who weigh less than 51 kg are classified as lightweight. Ren weighs 49 kg and she has a match against another randomly selected, lightweight woman boxer.
  2. Find the probability that Ren weighs less than the other boxer. In the tournament, women boxers who weigh more than \(H \mathrm {~kg}\) are classified as heavyweight. Given that \(10 \%\) of the women boxers in the tournament are classified as heavyweight,
  3. find the value of \(H\).
Edexcel S1 2024 January Q5
7 marks Standard +0.8
  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
Edexcel S1 2014 June Q4
5 marks Moderate -0.3
4. The random variable \(Y \sim \mathrm {~N} \left( \mu , \sigma ^ { 2 } \right)\) Given that \(\mathrm { P } ( Y < 17 ) = 0.6\) find
  1. \(\mathrm { P } ( Y > 17 )\)
  2. \(\mathrm { P } ( \mu < Y < 17 )\)
  3. \(\mathrm { P } ( Y < \mu \mid Y < 17 )\)
Edexcel S1 2022 June Q6
11 marks Standard +0.8
  1. A manufacturer fills bottles with oil. The volume of oil in a bottle, \(V \mathrm { ml }\), is normally distributed with \(V \sim \mathrm {~N} \left( 100,2.5 ^ { 2 } \right)\)
    1. Find \(\mathrm { P } ( V > 104.9 )\)
    2. In a pack of 150 bottles, find the expected number of bottles containing more than 104.9 ml
    3. Find the value of \(v\), to 2 decimal places, such that \(\mathrm { P } ( V > v \mid V < 104.9 ) = 0.2801\)
Edexcel S1 2013 January Q4
10 marks Standard +0.8
  1. The length of time, \(L\) hours, that a phone will work before it needs charging is normally distributed with a mean of 100 hours and a standard deviation of 15 hours.
    1. Find \(\mathrm { P } ( L > 127 )\).
    2. Find the value of \(d\) such that \(\mathrm { P } ( L < d ) = 0.10\)
    Alice is about to go on a 6 hour journey.
    Given that it is 127 hours since Alice last charged her phone,
  2. find the probability that her phone will not need charging before her journey is completed.
Edexcel S1 2014 June Q7
12 marks Moderate -0.3
7. The heights of adult females are normally distributed with mean 160 cm and standard deviation 8 cm .
  1. Find the probability that a randomly selected adult female has a height greater than 170 cm . Any adult female whose height is greater than 170 cm is defined as tall. An adult female is chosen at random. Given that she is tall,
  2. find the probability that she has a height greater than 180 cm . Half of tall adult females have a height greater than \(h \mathrm {~cm}\).
  3. Find the value of \(h\).
Edexcel S1 2015 June Q6
12 marks Moderate -0.3
  1. The random variable \(Z \sim \mathrm {~N} ( 0,1 )\) \(A\) is the event \(Z > 1.1\) \(B\) is the event \(Z > - 1.9\) \(C\) is the event \(- 1.5 < Z < 1.5\)
    1. Find
      1. \(\mathrm { P } ( A )\)
      2. \(\mathrm { P } ( B )\)
      3. \(\mathrm { P } ( C )\)
      4. \(\mathrm { P } ( A \cup C )\)
    The random variable \(X\) has a normal distribution with mean 21 and standard deviation 5
  2. Find the value of \(w\) such that \(\mathrm { P } ( X > w \mid X > 28 ) = 0.625\)
Edexcel S1 2017 June Q5
12 marks Standard +0.8
5. Yuto works in the quality control department of a large company. The time, \(T\) minutes, it takes Yuto to analyse a sample is normally distributed with mean 18 minutes and standard deviation 5 minutes.
  1. Find the probability that Yuto takes longer than 20 minutes to analyse the next sample. (3) The company has a large store of samples analysed by Yuto with the time taken for each analysis recorded. Serena is investigating the samples that took Yuto longer than 15 minutes to analyse. She selects, at random, one of the samples that took Yuto longer than 15 minutes to analyse.
  2. Find the probability that this sample took Yuto more than 20 minutes to analyse. Serena can identify, in advance, the samples that Yuto can analyse in under 15 minutes and in future she will assign these to someone else.
  3. Estimate the median time taken by Yuto to analyse samples in future.
Edexcel S1 Q4
13 marks Standard +0.3
4. Alan runs on a treadmill each day for as long as he can at 7 miles per hour. The length of time for which he runs is normally distributed with a mean of 21.6 minutes and a standard deviation of 1.8 minutes.
  1. Calculate the probability that on any one day Alan will run for less than 20 minutes.
  2. Estimate the number of times in a ninety-day period that Alan will run for more than 24 minutes.
  3. On a particular day Alan is still running after 22 minutes. Find the probability that he will stop running in the next 2 minutes.
Edexcel Paper 3 2018 June Q5
14 marks Challenging +1.2
  1. The lifetime, \(L\) hours, of a battery has a normal distribution with mean 18 hours and standard deviation 4 hours.
Alice's calculator requires 4 batteries and will stop working when any one battery reaches the end of its lifetime.
  1. Find the probability that a randomly selected battery will last for longer than 16 hours. At the start of her exams Alice put 4 new batteries in her calculator. She has used her calculator for 16 hours, but has another 4 hours of exams to sit.
  2. Find the probability that her calculator will not stop working for Alice's remaining exams. Alice only has 2 new batteries so, after the first 16 hours of her exams, although her calculator is still working, she randomly selects 2 of the batteries from her calculator and replaces these with the 2 new batteries.
  3. Show that the probability that her calculator will not stop working for the remainder of her exams is 0.199 to 3 significant figures. After her exams, Alice believed that the lifetime of the batteries was more than 18 hours. She took a random sample of 20 of these batteries and found that their mean lifetime was 19.2 hours.
  4. Stating your hypotheses clearly and using a \(5 \%\) level of significance, test Alice's belief.