Helen believes that the random variable C, representing cloud cover from the large data set, can be modelled by a discrete uniform distribution - Edexcel - A-Level Maths Mechanics - Question 1 - 2018 - Paper 2

To find the probability that cloud cover is less than 50%, let’s denote the maximum value of C as n. The cloud cover levels less than 50% would be the values from 0 up to \lfloor 0.5n \rfloor. Therefore, the probability can be calculated as:

$P(C < 0.5n) = \sum_{k=0}^{\lfloor 0.5n \rfloor} P(C=k) = \sum_{k=0}^{\lfloor 0.5n \rfloor} \frac{1}{n+1} = \frac{\lfloor 0.5n \rfloor + 1}{n+1}$

The model assumes a uniform distribution for cloud cover. However, the finding that 0.315 of days had cloud cover of less than 50% suggests that the distribution may not be uniform. The model would likely underestimate the actual hit rate of days with less than 50% cloud cover, indicating that it may not be suitable for representing real-world data.

A refinement could involve using a different distribution, such as a beta distribution, that allows for greater flexibility in modeling probabilities, especially in cases where certain outcomes are more likely than others. Additionally, incorporating historical data parameters to inform a more tailored probability distribution might yield better predictive accuracy.