Show that this price leaves you with a smaller profit than

Question

Question: You run a tutoring service for economics students. On half of all days the demand for your service is high, with

Q_H = 10 - 0.2P

Q_H is the number of hours you will work if you choose price P. On the remaining days, demand is low:

Q_L = 5 - 0.2P.

High and low days arrive at random with probabilities of 0.5 each. Your marginal opportunity cost is $10 per hour on all days. You want to choose a price that maximizes your average profit per day, but that price must be the same on all days. Find that price.

a. Show that this price leaves you with a smaller profit than if you can announce different prices early in the morning of each day, after you have learned whether demand will be high or low.