Sabrina counts the number of cars passing her house during randomly chosen one minute intervals. Two assumptions are needed for the number of cars passing her house in a fixed time interval to be well modelled by a Poisson distribution.
- State these two assumptions. [2]
- For each assumption in part (i) give a reason why it might not be a reasonable assumption for this context. [2]
Assume now that the number of cars that pass Sabrina's house in one minute can be well modelled by the distribution \(\text{Po}(0.8)\).
- Write down an expression for the probability that, in a given one minute period, exactly \(r\) cars pass Sabrina's house. [1]
- Hence find the probability that, in a given one minute period, exactly 2 cars pass Sabrina's house. [1]
- Find the probability that, in a given 30 minute period, at least 28 cars pass Sabrina's house. [3]
- The number of bicycles that pass Sabrina's house in a 5 minute period is a random variable with the distribution \(\text{Po}(1.5)\). Find the probability that, in a given 10 minute period, the total number of cars and bicycles that pass Sabrina's house is between 12 and 15 inclusive. State a necessary condition. [4]