A machine is used to bag coal, the mass of coal delivered per bag being normally distributed with mean 55 Kg and standard deviation 1.25 Kg.
Given two filled bags chosen at random calculate the probabilities that
(i) each bag contains at least 56 Kg (ii) one bag contains at least 1 Kg more than the other bag
Answers: (i) 0.045 and (ii) 0.572
For (i) the combined mean of the 2 bags is 112 Kg and the standard deviation is 1.25 + 1.25 = 2.50
For (ii) Let one bag be X and the other Y so we have mod(X-Y)>1. The mean is E(X) - E(Y) = 0 and we have Var(X) + Var(Y) = 3.125, so we have a standard deviation of sqrt(3.125).
Hints
For (i), you have both $X,Y \sim \mathcal{N}(55, 1.25)$ and $X,Y$ are independent.
Then, $$ \mathbb{P}[X > 56, Y > 56] = \mathbb{P}[X > 56] \times \mathbb{P}[Y > 56] $$ but $X,Y$ have the same distribution, can you take it from here?
For (ii), look at $\mathbb{P}[|X-Y| > 1]$ using the joint distribution between $X$ and $Y$, or condition on $Y$. Note that $|X-Y| > 1$ means either $X-Y>1$ or $Y-X>1$...