# Probability that a draw from a normal distribution is some number greater than another draw from the same distribution

I hope to learn the general way of solving this problem, but I have this specific problem: $$X\sim N(\mu,\sigma^2) \\ \mu=470,\ \sigma=70$$ If two people, A and B, each draw one entry from the same distribution, what is the probability that A's entry is at least 100 more than B's entry?

• Let $A$ and $B$ be two independent random variables both with distribution $\mathcal{N}(\mu, \sigma^2)$. What is the distribution of the random variable $A-B$? Apr 3, 2019 at 13:07
• You have $A-B>100$. Rearrange to $A-B-100>0$. Now all you have to do is figuring out the distribution of $A-B-100$. In this case, this is quite easy (hint: it's still a normal distribution). Apr 3, 2019 at 13:08
• Hint: $A-B$ is also a normal random variable. Can you figure out what the mean and variance of $A-B$ are? If so, you should be able to answer the question; What is the probability that $A-B$ is $100$ or more? Apr 3, 2019 at 13:10

Since it looks like self-study question, I'll start with a hint: Think of $$X_1-X_2$$ with $$X_1, X_2 \sim N(470, 70^2)$$. What distribution does $$X_1-X_2$$ follow? How to interpret $$X_1-X_2$$?