The question goes as follows:

A shoe factory produces brown shoes and black shoes. They look the same but differ only in their weight characteristics. Brown shoes have their weight distributed as Normal$(\mu = 7 \space\text{lbs}, \sigma^2)$ and black shoes as Normal$(\mu = 8 \space\text{lbs}, \sigma^2)$. They are produced in equal proportions and the black shoes are, on average, 1 lbs heavier than brown shoes.

a) Write an equation for the prior predictive distribution of the weight of a randomly selected pair of shoes.

b) You are provided a shoe, and find that it weighs 7 lbs. Find the conditional probability that it's a brown shoe, when $\sigma = 1$.

c) Explain how the probability that a 7 lbs shoe is brown will vary as $\sigma$ increases and decreases.

I am mostly interested in parts (a) and (b), as I think for (c), the answer is that as $\sigma$ goes down, the probability that a shoe is brown will increase and vice versa.

For (a), my initial thoughts are that it would be $\text{Normal}(\mu = 7.5, 2\sigma^2)$ as I think the variances will add. However, I'm not quite sure of whether this is the correct mean.

Fort part (b), I'm really confused and would appreciate any detailed help. Many thanks!