Poisson with geometric or waiting time

A question is this type if and only if it involves finding probabilities related to waiting times or the number of trials until a Poisson event occurs.

4 questions · Moderate -0.5

5.02i Poisson distribution: random events model
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Edexcel S2 2014 June Q1
8 marks Moderate -0.8
  1. Patients arrive at a hospital accident and emergency department at random at a rate of 6 per hour.
    1. Find the probability that, during any 90 minute period, the number of patients arriving at the hospital accident and emergency department is
      1. exactly 7
      2. at least 10
    A patient arrives at 11.30 a.m.
  2. Find the probability that the next patient arrives before 11.45 a.m.
AQA Further Paper 3 Statistics 2023 June Q2
1 marks Moderate -0.5
2 The time, \(T\) days, between rain showers in a city in autumn can be modelled by an exponential distribution with mean 1.25 Find the distribution of the number of rain showers per day in the city.
Tick ( \(\checkmark\) ) one box.
[0pt] [1 mark] \includegraphics[max width=\textwidth, alt={}, center]{1e2fdd33-afa4-486f-a9e2-1d425ed14eee-03_108_113_1800_370}
DistributionMean
Exponential0.8
\includegraphics[max width=\textwidth, alt={}]{1e2fdd33-afa4-486f-a9e2-1d425ed14eee-03_108_113_1932_370}
Exponential1.25
□
Poisson0.8
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Poisson1.25
WJEC Unit 2 2024 June Q2
10 marks Moderate -0.8
A baker sells 3-5 birthday cakes per hour on average.
  1. State, in context, two assumptions you would have to make in order to model the number of birthday cakes sold using a Poisson distribution. [1]
  2. Using a Poisson distribution and showing your calculation, find the probability that exactly 2 birthday cakes are sold in a randomly selected 1-hour period. [2]
  3. Calculate the probability that, during a randomly selected 3-hour period, the baker sells more than 10 birthday cakes. [3]
  4. The baker sells a birthday cake at 9:30 a.m. Calculate the probability that the baker will sell the next birthday cake before 10:00 a.m. [3]
  5. Select one of the assumptions in part (a) and comment on its reasonableness. [1]
WJEC Further Unit 2 Specimen Q6
10 marks Standard +0.3
Customers arrive at a shop such that the number of arrivals in a time interval of \(t\) minutes follows a Poisson distribution with mean \(0.5t\).
  1. Find the probability that exactly 5 customers arrive between 11 a.m. and 11.15 a.m. [3]
  2. A customer arrives at exactly 11 a.m.
    1. Let the next customer arrive at \(T\) minutes past 11 a.m. Show that $$P(T > t) = e^{-0.5t}.$$
    2. Hence find the probability density function, \(f(t)\), of \(T\).
    3. Hence, giving a reason, write down the mean and the standard deviation of the time between the arrivals of successive customers. [7]