Probability of ordered outcome between components

A question is this type if and only if it asks for the probability that one Poisson component exceeds (or is less than) another component, such as more goals in one half than the other, requiring summation over valid combinations.

2 questions · Standard +0.6

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 2020 June Q5
10 marks Standard +0.3
5 Each week a sports team plays one home match and one away match. In their home matches they score goals at a constant average rate of 2.1 goals per match. In their away matches they score goals at a constant average rate of 0.8 goals per match. You may assume that goals are scored at random times and independently of one another.
  1. A week is chosen at random.
    1. Find the probability that the team scores a total of 4 goals in their two matches.
    2. Find the probability that the team scores a total of 4 goals, with more goals scored in the home match than in the away match.
  2. Use a suitable approximating distribution to find the probability that the team scores fewer than 25 goals in 10 randomly chosen weeks.
  3. Justify the use of the approximating distribution used in part (b).
CAIE S2 2013 November Q4
9 marks Standard +0.8
4 Goals scored by Femchester United occur at random with a constant average of 1.2 goals per match. Goals scored against Femchester United occur independently and at random with a constant average of 0.9 goals per match.
  1. Find the probability that in a randomly chosen match involving Femchester,
    1. a total of 3 goals are scored,
    2. a total of 3 goals are scored and Femchester wins. The manager promises the Femchester players a bonus if they score at least 35 goals in the next 25 matches.
    3. Find the probability that the players receive the bonus.