The owner of a ski apparel store in Winter Park, Colorado, must make a decision in July regarding the number of ski jackets to order for the following ski season. Each ski jacket costs $$54$ and can be sold during the ski season for $$145.$ Any unsold jackets at the end of the season are sold for $$45.$ The demand for jackets is expected to follow the Poisson distribution with an average rate of $80.$ The store owner can order jackets in lot sizes of $10$ units.

$(1)$ How many jackets should the store owner order to maximize expected profit?

$(2)$ When maximizing expected profit, what are the best$-$case and worst$-$case outcomes?

$(3)$ When maximizing expected profit, how likely is it that the store owner will make at least $$7,000?$

$(4)$ When maximizing expected profit, how likely is it that the store owner will make betwen $$6,000$ to $$7,000?$

Check figures: $(1)$ Order quant. $=90$

$(2)$ Best case about $8,190$

$(3)$ About $57\%$

$(4)$ About $33\%$