Rescale rate then sum Poissons

A question is this type if and only if the two (or more) Poisson processes have rates given in different time or space units that must first be rescaled to a common unit before summing and computing a probability.

6 questions · Standard +0.3

5.02i Poisson distribution: random events model5.02j Poisson formula: P(X=x) = e^(-lambda)*lambda^x/x!5.02k Calculate Poisson probabilities
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CAIE S2 2003 June Q6
10 marks Standard +0.3
6 Computer breakdowns occur randomly on average once every 48 hours of use.
  1. Calculate the probability that there will be fewer than 4 breakdowns in 60 hours of use.
  2. Find the probability that the number of breakdowns in one year (8760 hours) of use is more than 200.
  3. Independently of the computer breaking down, the computer operator receives phone calls randomly on average twice in every 24 -hour period. Find the probability that the total number of phone calls and computer breakdowns in a 60-hour period is exactly 4 .
CAIE S2 2022 June Q5
10 marks Standard +0.3
5 Cars arrive at a fuel station at random and at a constant average rate of 13.5 per hour.
  1. Find the probability that more than 4 cars arrive during a 20-minute period.
  2. Use an approximating distribution to find the probability that the number of cars that arrive during a 12-hour period is between 150 and 160 inclusive.
    Independently of cars, trucks arrive at the fuel station at random and at a constant average rate of 3.6 per 15-minute period.
  3. Find the probability that the total number of cars and trucks arriving at the fuel station during a 10 -minute period is more than 3 and less than 7 .
CAIE S2 2017 June Q6
9 marks Standard +0.3
6 Old televisions arrive randomly and independently at a recycling centre at an average rate of 1.2 per day.
  1. Find the probability that exactly 2 televisions arrive in a 2-day period.
  2. Use an appropriate approximating distribution to find the probability that at least 55 televisions arrive in a 50-day period.
    Independently of televisions, old computers arrive randomly and independently at the same recycling centre at an average rate of 4 per 7-day week.
  3. Find the probability that the total number of televisions and computers that arrive at the recycling centre in a 3-day period is less than 4.
CAIE S2 2010 November Q2
5 marks Standard +0.3
2 People arrive randomly and independently at a supermarket checkout at an average rate of 2 people every 3 minutes.
  1. Find the probability that exactly 4 people arrive in a 5 -minute period. At another checkout in the same supermarket, people arrive randomly and independently at an average rate of 1 person each minute.
  2. Find the probability that a total of fewer than 3 people arrive at the two checkouts in a 3 -minute period.
AQA S2 2015 June Q1
9 marks Standard +0.3
1 In a survey of the tideline along a beach, plastic bottles were found at a constant average rate of 280 per kilometre, and drinks cans were found at a constant average rate of 220 per kilometre. It may be assumed that these objects were distributed randomly and independently. Calculate the probability that:
  1. a 10 m length of tideline along this beach contains no more than 5 plastic bottles;
  2. a 20 m length of tideline along this beach contains exactly 2 drinks cans;
  3. a 30 m length of tideline along this beach contains a total of at least 12 but fewer than 18 of these two types of object.
    [0pt] [4 marks]
WJEC Further Unit 2 2024 June Q1
14 marks Standard +0.3
  1. Dave and Llinos like to go fishing. When they go fishing, on average, Dave catches 4.3 fish per day and Llinos catches 3.8 fish per day. A day of fishing is assumed to be 8 hours.
    1. (i) Calculate the probability that they will catch fewer than 2 fish in total on a randomly selected half-day of fishing.
      (ii) Justify any distribution you have used in answering (a)(i).
    2. On a randomly selected day, Dave starts fishing at 7 am. Given that Dave has not caught a fish by 11 am,
      1. find the expected time he catches his first fish,
      2. calculate the probability that he will not catch a fish by 3 pm .
    3. On average, only \(2 \%\) of the fish that Llinos catches are trout. Over the course of a year, she catches 950 fish. Calculate the probability that at least 30 of these fish are trout. [3]
      [0pt] she catches 950 fish. Calculate the probability that at least 30 of these fish are trout. [3]
    4. State, with a reason, a distribution, including any parameters, that could approximate the distribution used in part (c).
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