Conditional or competitive scenarios

Questions involving conditional probabilities or competitive outcomes between processes (e.g., one team scoring more than another, overflow between guesthouses), requiring consideration of relative outcomes rather than just joint or total counts.

2 questions

CAIE S2 2013 November Q4
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,
    (a) a total of 3 goals are scored,
    (b) 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.
  2. Find the probability that the players receive the bonus.
OCR S3 2015 June Q5
5 Two guesthouses, the Albion and the Blighty, have 8 and 6 rooms respectively. The demand for rooms at the Albion has a Poisson distribution with mean 6.5 and the demand for rooms at the Blighty has an independent Poisson distribution with mean 5.5. The owners have agreed that if their guesthouse is full, they will re-direct guests to the other.
  1. Find the probability that, on any particular night, the two guesthouses together do not have enough rooms to meet demand.
  2. The Albion charges \(\pounds 60\) per room per night, and the Blighty \(\pounds 80\). Find the probability, that on a particular night, the total income of the two guesthouses is exactly \(\pounds 400\).
  3. If \(A\) is the number of rooms demanded at the Albion each night, and \(B\) the number of rooms demanded at the Blighty each night, find the mean and variance of the variable \(C = 60 A + 80 B\). State whether \(C\) has a Poisson distribution, giving a reason for your answer.