Conditional probability given total

A question is this type if and only if it asks for the probability of a specific split between two Poisson components given that their combined total equals a fixed value.

3 questions · Standard +0.3

5.02i Poisson distribution: random events model5.02j Poisson formula: P(X=x) = e^(-lambda)*lambda^x/x!
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CAIE S2 2024 June Q5
9 marks Standard +0.3
5 The number of goals scored by a sports team in the first half of any match has the distribution \(X \sim \mathrm { Po }\) (3.1). The number of goals scored by the same team in the second half of any match has the distribution \(Y \sim \operatorname { Po } ( 2.4 )\). You may assume that the distributions of \(X\) and \(Y\) are independent.
  1. Find \(\mathrm { P } ( X < 4 )\).
  2. Find the probability that, in a randomly chosen match, the team scores at least 5 goals.
  3. Given that the team scores a total of 5 goals in a randomly chosen match, find the probability that they score exactly 3 goals in the first half.
Edexcel S2 2018 January Q5
15 marks Standard +0.3
5. A delivery company loses packages randomly at a mean rate of 10 per month. The probability that the delivery company loses more than 12 packages in a randomly selected month is \(p\)
  1. Find the value of \(p\) The probability that the delivery company loses more than \(k\) packages in a randomly selected month is at least \(2 p\)
  2. Find the largest possible value of \(k\) In a randomly selected month,
  3. find the probability that exactly 4 packages were lost in each half of the month. In a randomly selected two-month period, 21 packages were lost.
  4. Find the probability that at least 10 packages were lost in each of these two months.
  5. Using a suitable approximation, find the probability that more than 27 packages are lost during a randomly selected 4-month period.
Edexcel FS1 AS 2020 June Q4
8 marks Standard +0.3
  1. During the morning, the number of cyclists passing a particular point on a cycle path in a 10-minute interval travelling eastbound can be modelled by a Poisson distribution with mean 8
The number of cyclists passing the same point in a 10 -minute interval travelling westbound can be modelled by a Poisson distribution with mean 3
  1. Suggest a model for the total number of cyclists passing the point on the cycle path in a 10-minute interval, stating a necessary assumption. Given that exactly 12 cyclists pass the point in a 10 -minute interval,
  2. find the probability that at least 11 are travelling eastbound. After some roadworks were completed, the total number of cyclists passing the point in a randomly selected 20-minute interval one morning is found to be 14
  3. Test, at the \(5 \%\) level of significance, whether there is evidence of a decrease in the rate of cyclists passing the point.
    State your hypotheses clearly.