I have a box of small components and a box of large components. The mean of the small components is less than the mean of the large ones, but the distributions overlap.
Question: If I pick one component from each box at random, what is the probability that the component from the box of "small" parts will be larger than the one from the box of "large" parts?
This is the method I'm planning to try (in Excel), is it correct? Is there an easier way?
1. tabulate an incrementing series of sizes.
2. from the mean and standard deviation, calculate the probability of the small component being equal to each size tabulated
3. calculate the probability of the large component being less than the size above
4. multiply 2 & 3 to get the probability of both simultaneously
5. use simpsons rule to integrate the area of the product in 4 above for all sizes in the series (assuming that the table covers a range extending beyond several standard deviations)
The alternative I'm thinking of is to use the probability of the small component being larger than each size tabulated in step 2 above, but that intuitively feels like I would be counting events multiple times.