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 values of C less than 4, which are {0, 1, 2, 3}.

The probability can be calculated as:

$P(C < 4) = P(C = 0) + P(C = 1) + P(C = 2) + P(C = 3) = 4 \times \frac{1}{9} = \frac{4}{9} \approx 0.444$

Thus, the probability that cloud cover is less than 50% is approximately 0.444.

Considering that the proportion of days with cloud cover of less than 50% from data is 0.315, and from our model it is approximately 0.444, we can infer that Helen's discrete uniform distribution model does not accurately represent the observed data.

The model indicates a higher probability for less than 50% cloud cover than the actual data suggests, which indicates that the uniform distribution is not suitable.

A suitable refinement to Helen's model could be to develop a non-uniform distribution. This accounts for variations in cloud cover based on factors such as seasonal changes or geographical locations. Instead of assuming equal probabilities for all values of C, using a model that reflects the actual observed frequencies of cloud cover percentages can produce more accurate predictions.