Question

At a certain coffee​ shop, all the customers buy a cup of coffee and some also buy a doughnut. The shop owner believes that the number of cups he sells each day is normally distributed with a mean of 310 cups and a standard deviation of 25 cups. He also believes that the number of doughnuts he sells each day is independent of the coffee sales and is normally distributed with a mean of 160 doughnuts and a standard deviation of 17

a) The shop is open every day but Sunday. Assuming​ day-to-day sales are​ independent, what's the probability​ he'll sell over 2000 cups of coffee in a​ week?

b) If he makes a profit of 50 cents of each cup of coffee and 40 cents on each doughnut, can he reasonably expect to have a day's profit of over $300? Explain.

c) What's the probability that on any given day he'll sell a doughnut to more than half of his coffee customers?

Answer 1

Let X denote the number of cups sold each day and Y denote the number of doughnuts sold each day. Then,

a)

Using Central limit theorem, we know

Required probability =

b)

Now,

So,

which is quite low. So, it is highly unlikely that he would earn a day's profit of over $300.

c)

Now,

Required probability =