Poisson over different time periods, conditional probability given a total, non-constant rates
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A district council models the number of landslips closing a rural road in each winter month (June, July, August) as Poisson with mean 1.5, independently from month to month.
Figure 1 shows the number of road-closing landslips in each of the last 20 winters (2006–2025) (illustrative council records). Over these 20 winters there were 90 landslips in total, a mean of 4.5 per winter, which is where the model's mean of 1.5 per month comes from.
Bar graph of the number of landslips closing the road in each winter from 2006 to 2025 (illustrative). Counts vary from 1 to 10 per winter, with the four most recent winters all having 7 or more.Facts you may need
(a)
Find the probability of 7 or more landslips over the three winter months, and the probability of no landslips in June.
(b)
Given there were at most 2 landslips over the whole winter, find the probability that June had none. Explain why this differs from P(June=0).
(c)
Show that, given exactly 2 landslips in the winter, the probability both occurred in June is 1/9, and explain why this makes sense.
(d)
Use Figure 1 and the context to discuss TWO reasons why a Poisson model with the same mean each month may not be appropriate, and the effect on the probability in (a).
