I was solving an exam problem:
Website's number of clicks per month is a random variable with normal distribution, where the mean is 12 and the variance is 2 (both in millions). Calculate the probability that 15 000 000 clicks per month occur.
Here is the catch: there is a difference between a normal distribution with the mean of 12 000 000 and variance of 2 000 000, and a normal distribution with the mean of 12 and the variance of 2.
If I went along with units in millions, so a normal distribution with the mean of 12 and the variance of 2, this would be my result using transformed variables: $$U = \frac{X-\mu}{\sigma}$$ where $\mu=12$ and $\sigma = \sqrt{2}$
$$1-\Phi(\frac{15-12}{\sqrt{2}})=0.016947$$
If I went along with bare numbers, this would be my result:
$$U = \frac{X-\mu}{\sigma}$$ where $\mu=12 000 000$ and $\sigma = \sqrt{2000000}$
$$1-\Phi(\frac{15000000-12000000}{\sqrt{2000000}})=0.00000000001$$
The first result was correct, and I got it right, but nonetheless I think that the problem was stated confusingly and ambiguously. I know why the difference happens, all I'm asking is: What should my intuition be in cases like this? I wrote the problem verbatim, they really did mix units in millions and bare units. What is your take on this?