Normal distribution tricky problem

Question

The question goes like this:

The amount of regular petrol purchased every week at a petrol pump in a city follows the normal distribution with mean 50000 litres and standard deviation 10000 litres. The starting supply of petrol is 74000 litres, and there is scheduled weekly delivery of 47000 litres. Find the probability that, after 11 weeks, the supply of petrol will be below 20000 gallons.

Without that weekly information the problem is easy. But, with that information I am not sure which quantity changes. I believe mean must change, but I am not sure about the standard deviation.

Answer 1

The change of supply from week to week is normally distributed, with mean -3000 litres, and a standard deviation of 10000 litres.

The sum of $11$ independent normal variables all with mean $m$ and standard deviation $s$ is again a normal variable with mean $11m$ and standard deviation $\sqrt{11}s$ (this is also true for any other number in place of $11$). Assuming the tank is allowed to go into negative supply (or, more likely, the restocking can be expedited if needed), you can tell from there how much the supply has changed with what probability.

Answer 2

First, take into account the things that are constant: starting with $74000 L$ and gaining $47000 L$ $11$ times.

$$74000 + 11*47000 = 591000 L$$

In order for the station to have $20000 L$ or less, they need to sell at least $571000 L$ over those $11$ weeks. Based on the information we were given, we expect that they will sell $11*50000 = 550000 L$ in $11$ weeks with a standard deviation of $11*10000 = 110000 L$.

From this, we calculate a $Z-$score of about $.19$. Thus the probability of this happening is about $.4247$.