1. 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 1

To find the probability that cloud cover is less than 50%, we consider the relevant values for C:

From the distribution, cloud cover less than 50% corresponds to C = 0, 1, 2, 3. Therefore, there are 4 favorable outcomes (0 to 3) out of a total of 9 possible outcomes.

Thus, the probability is given by:

$P(C < 4) = \frac{4}{9} \approx 0.444$

This indicates that the probability of cloud cover being less than 50% is approximately 44.4%.

Helen’s model predicts a probability of $\frac{4}{9}$, or approximately 0.444, for cloud cover being less than 50%. However, her analysis of the historical data indicates that the actual proportion of days with cloud cover of less than 50% is 0.315.

Given that this observed proportion is significantly lower than the predicted probability, it suggests that the discrete uniform distribution is not an accurate model for the cloud cover data. It indicates that cloud cover does not vary uniformly across the different values.

Therefore, the model fails to account for the variability and may misrepresent the actual distribution of cloud cover.

A suitable refinement to Helen's model would be to consider a non-uniform distribution, which can better account for fluctuations in cloud cover based on various factors such as time of year, geographic location, and seasonal weather patterns.

For instance, employing a bi-modal or skewed distribution could capture scenarios where there are more days with either low or high cloud cover, rather than being uniformly distributed across all levels. This would likely lead to a more accurate representation of cloud cover variability.