Conditional probability with normal

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

8 questions

Edexcel S1 2016 January Q5
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
  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 2014 June Q4
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
  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
  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
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 2017 June Q5
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
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.