A merchant stocks a perishable item. He knows that on any given day he will have demand for two, three, or four of these items, with probabilities .2, .3, and .5. He buys items for $1 and sells them for $1.20. Any items left over at the end of the day represent a loss. How many items should the merchant stock to maximize his expected daily profit?

Work:

I think I need to use Tchebysheff's theorem but I'm unsure how to incorporate prices etc. into the problem.