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Suppose you want to plot the probability distribution that you will make money manufacturing widgets. You have a table of the probabilities associated with the various prices you might be able to negotiate from a widget-component supplier; you have a table of the probabilities associated with the costs of assembling the widgets; and you have a table of the probability of the prices that you might be able to sell the widgets for.

I presume one way to do it would be to proceed by brute force and multiply the probabilities for each possible combination. Is there a more efficient way to compute the answers?

Practically speaking, is there a way this can be done more efficiently using a spreadsheet program?

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You are describing three random variables ($C_{supplier}$, $C_{assembly}$, $P$), and a fourth one that is profit $\pi = P - C_{supplier} - C_{assembly}$.

If you can reasonably approximate all of these variables with Normal distributions, and reasonably assume they are independent, then a reasonable estimate of the mean (variance) of profit is the mean (variance) of $P$ minus the means (variances) of $C_{supplier}$ and $C_{assembly}$:

https://en.wikipedia.org/wiki/Sum_of_normally_distributed_random_variables

If the Normal distribution is a bad fit, you can choose a different distribution and Google the rules for subtracting those random variables.

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  • $\begingroup$ In other words, your answer is "Google the answer!" $\endgroup$ – whuber Mar 10 at 15:25
  • $\begingroup$ The title of the question indicates OP is interested in multinomial distributions with exactly six outcomes each (i.e., dice rolls), so I was expecting the question to get marked as a duplicate for some other "sums of multinomials" question, or OP to get scolded for asking something that sounds like homework. But the body suggests they are not necessarly interested in just six outcomes, so I wanted to help them structure their thoughts. $\endgroup$ – suckrates Mar 10 at 15:38
  • $\begingroup$ Fair enough--it's well-intentioned and everyone appreciates that. But helping to structure thoughts perhaps is better done through comments to the question. $\endgroup$ – whuber Mar 10 at 15:39
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    $\begingroup$ Wasn't for homework. It was a problem that came up at work. Didn't know the phrases for effrctigr googling of the answer. Thank you for the help. $\endgroup$ – Hal Mar 22 at 17:39

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