# Question about normal distribution curves

I’m in AP Statistics, and a question came up that is about two normal distributions. One is the height of men, with a mean of 69.5 and a standard deviation of 4, and the other is the height of women, with a mean of 65 and a standard deviation of 3. Is there a way I can overlap the two distributions and look at their areas to calculate the probability a man is taller than a woman? I’m not sure how to do it and I’m wondering if it is possible.

Let $$X$$ be a random variable representing a man's height and let $$Y$$ be a random variable representing a woman's height.
Now let $$Z = X - Y$$ or the difference in height between the man and the woman. You are effectively asking $$P(X>Y)$$ which is the same as asking $$P(X-Y>0)$$ or $$P(Z>0)$$.
The "normal difference distribution" (i.e. the distribution of $$X-Y$$) where $$X \sim N(69.5, 4^2)$$ and $$Y \sim N(65, 3^2)$$ is normally distributed as $$N(69.5-65, 4^2+3^2)$$ or $$N(4.5, 25)$$. You can integrate over the Normal PDF from $$0$$ to $$\infty$$ now in order to find $$P(X-Y>0)$$.
Overlaying the PDFs and integrating will tell you things like "there is a 60% chance that men are between $$x$$ and $$y$$ feet tall while there is only a 30% chance a woman will be between $$x$$ and $$y$$ feet tall", but it won't tell you anything about the probability a man will actually be taller than a woman. To figure that out, you need the joint distribution of $$X$$ and $$Y$$ or you need to find the distribution of $$X-Y$$, which I did above.
• worth mentioning is the assumption of zero covariance between $X$ and $Y$