Conditional probability with Poisson

A question is this type if and only if it asks for P(A|B) where both events involve Poisson random variables, requiring use of conditional probability formula.

8 questions

CAIE S2 2020 June Q4
4 The random variable \(A\) has the distribution \(\operatorname { Po } ( 1.5 ) . A _ { 1 }\) and \(A _ { 2 }\) are independent values of \(A\).
  1. Find \(\mathrm { P } \left( A _ { 1 } + A _ { 2 } < 2 \right)\).
  2. Given that \(A _ { 1 } + A _ { 2 } < 2\), find \(\mathrm { P } \left( A _ { 1 } = 1 \right)\).
  3. Give a reason why \(A _ { 1 } - A _ { 2 }\) cannot have a Poisson distribution.
CAIE S2 2024 June Q7
7 The independent random variables \(X\) and \(Y\) have the distributions \(\operatorname { Po } ( 1.9 )\) and \(\operatorname { Po } ( 2.2 )\) respectively.
  1. Find \(\mathrm { P } ( X + Y < 4 )\).
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    \includegraphics[max width=\textwidth, alt={}, center]{7c078a14-98f9-4292-ae76-a2642238176f-10_75_1581_497_322}
  2. Find the probability that \(X = 2\) given that \(X + Y < 4\).
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  3. A sample of 60 randomly chosen pairs of values of \(X\) and \(Y\) is taken,and the value of \(X + Y\) is calculated for each pair.The sample mean of these 60 values is found. Find the probability that the sample mean of \(X + Y\) is less than 4.0 .
    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 S2 2018 June Q4
4 The numbers, \(M\) and \(F\), of male and female students who leave a particular school each year to study engineering have means 3.1 and 0.8 respectively.
  1. State, in context, one condition required for \(M\) to have a Poisson distribution.
    Assume that \(M\) and \(F\) can be modelled by independent Poisson distributions.
  2. Find the probability that the total number of students who leave to study engineering in a particular year is more than 3 .
  3. Given that the total number of students who leave to study engineering in a particular year is more than 3 , find the probability that no female students leave to study engineering in that year.
CAIE S2 2004 June Q6
6 At a certain airfield planes land at random times at a constant average rate of one every 10 minutes.
  1. Find the probability that exactly 5 planes will land in a period of one hour.
  2. Find the probability that at least 2 planes will land in a period of 16 minutes.
  3. Given that 5 planes landed in an hour, calculate the conditional probability that 1 plane landed in the first half hour and 4 in the second half hour.
CAIE S2 2013 June Q4
4 The independent random variables \(X\) and \(Y\) have the distributions \(\operatorname { Po } ( 2 )\) and \(\operatorname { Po } ( 3 )\) respectively.
  1. Given that \(X + Y = 5\), find the probability that \(X = 1\) and \(Y = 4\).
  2. Given that \(\mathrm { P } ( X = r ) = \frac { 2 } { 3 } \mathrm { P } ( X = 0 )\), show that \(3 \times 2 ^ { r - 1 } = r\) ! and verify that \(r = 4\) satisfies this equation.
Edexcel S2 2023 June Q7
  1. A bakery sells muffins individually at an average rate of 8 muffins per hour.
    1. Find the probability that, in a randomly selected one-hour period, the bakery sells at least 4 but not more than 8 muffins.
    A sample of 5 non-overlapping half-hour periods is selected at random.
  2. Find the probability that the bakery sells fewer than 3 muffins in exactly 2 of these periods. Given that 4 muffins were sold in a one-hour period,
  3. find the probability that more muffins were sold in the first 15 minutes than in the last 45 minutes.
Edexcel S2 2018 Specimen Q1
  1. The number of cars caught speeding per day, by a particular camera, has a Poisson distribution with mean 0.8
    1. Find the probability that in a given 4 day period exactly 3 cars will be caught speeding by this camera.
    A car has been caught speeding by this camera.
  2. Find the probability that the period of time that elapses before the next car is caught speeding by this camera is less than 48 hours. Given that 4 cars were caught speeding by this camera in a two day period,
  3. find the probability that 1 was caught on the first day and 3 were caught on the second day. Each car that is caught speeding by this camera is fined £60
  4. Using a suitable approximation, find the probability that, in 90 days, the total amount of fines issued will be more than \(\pounds 5000\)
OCR Further Statistics AS 2024 June Q4
  1. Find the probability that 4 telephone calls are received in a randomly chosen one-minute period.
  2. A sample of 10 independent observations of \(X\) is obtained. Find the expected number of these 10 observations that are in the interval \(2 < X < 8\). It is also known that
    \(P ( X + Y = 4 ) = \frac { 27 } { 8 } P ( X = 2 ) \times P ( Y = 2 )\).
  3. Determine the possible values of \(\mathrm { E } ( Y )\).
  4. Explain where in your solution to part (c) you have used the assumption that telephone calls and e-mails are received independently of one another.