Combined events across distributions

Questions involving probability of combined events from multiple normal distributions, such as both/all satisfying conditions simultaneously, or selecting from multiple populations (e.g., probability both a child and adult meet criteria, exactly one of several meeting a condition).

5 questions

CAIE S1 2022 March Q4
4 The weights of male leopards in a particular region are normally distributed with mean 55 kg and standard deviation 6 kg .
  1. Find the probability that a randomly chosen male leopard from this region weighs between 46 and 62 kg .
    The weights of female leopards in this region are normally distributed with mean 42 kg and standard deviation \(\sigma \mathrm { kg }\). It is known that \(25 \%\) of female leopards in the region weigh less than 36 kg .
  2. Find the value of \(\sigma\).
    The distributions of the weights of male and female leopards are independent of each other. A male leopard and a female leopard are each chosen at random.
  3. Find the probability that both the weights of these leopards are less than 46 kg .
CAIE S1 2008 June Q4
4 In a certain country the time taken for a common infection to clear up is normally distributed with mean \(\mu\) days and standard deviation 2.6 days. \(25 \%\) of these infections clear up in less than 7 days.
  1. Find the value of \(\mu\). In another country the standard deviation of the time taken for the infection to clear up is the same as in part (i), but the mean is 6.5 days. The time taken is normally distributed.
  2. Find the probability that, in a randomly chosen case from this country, the infection takes longer than 6.2 days to clear up.
OCR MEI S2 2008 January Q3
3 In a large population, the diastolic blood pressure (DBP) of 5-year-old children is Normally distributed with mean 56 and standard deviation 6.5.
  1. Find the probability that the DBP of a randomly selected 5-year-old child is between 52.5 and 57.5. The DBP of young adults is Normally distributed with mean 68 and standard deviation 10.
  2. A 5-year-old child and a young adult are selected at random. Find the probability that the DBP of one of them is over 62 and the other is under 62.
  3. Sketch both distributions on a single diagram.
  4. For another age group, the DBP is Normally distributed with mean 82. The DBP of \(12 \%\) of people in this age group is below 62. Find the standard deviation for this age group.
Edexcel S1 2006 January Q7
7. The heights of a group of athletes are modelled by a normal distribution with mean 180 cm and a standard deviation 5.2 cm . The weights of this group of athletes are modelled by a normal distribution with mean 85 kg and standard deviation 7.1 kg . Find the probability that a randomly chosen athlete
  1. is taller than 188 cm ,
  2. weighs less than 97 kg .
    (2)
  3. Assuming that for these athletes height and weight are independent, find the probability that a randomly chosen athlete is taller than 188 cm and weighs more than 97 kg .
  4. Comment on the assumption that height and weight are independent.
SPS SPS FM Statistics 2021 September Q5
5. The heights of a population of men are normally distributed with mean \(\mu \mathrm { cm }\) and standard deviation \(\sigma \mathrm { cm }\). It is known that \(20 \%\) of the men are taller than 180 cm and \(5 \%\) are shorter than 170 cm .
a Sketch a diagram to show the distribution of heights represented by this information.
b Find the value of \(\mu\) and \(\sigma\).
c Three men are selected at random, find the probability that they are all taller than 175 cm .
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