Hello, I am stuck at this problem relating to probability.

Sarah sells Valetine's Day theme keychains on Etsy. The estimated demand during the selling period is normally distributed with a mean of 29 items and a standard deviation of 17 items. Sarah sells the keychains at $25 per unit and it costs her $5 per unit. If Sarah decides to set the order quantity of Valentine's day keychain to be 31:

1. What would be the expected # of units short under this policy?
2. What would be the item fill rate under this policy?

Sarah has realized that her order quantity is not optimal and wants to calculate the optimal value in order to maximize profit.

1. How many keychains should Sarah order in October to maximize profit?

Sarah has hired a supply chain specialist to cross-check the order quantity that was proposed above. The supply chain specialist suggested to use an order quantity of 52. To show that your order quantity is truly the optimal value, you decided to calculate the expected profit for the order quantity that you proposed (in Question 2); and the expected profit for the order quantity recommended by the supply chain specialist.

1. What is the expected profit using the order quantity that you proposed in Question 2?
2. What is the expected profit using the order Q proposed by the SC specialist?

After doing some market research, Sarah discovers that she could sell the leftover inventory of keychains after Valentine's day if the selling price is drastically reduced. She decides that she will be selling the remaining keychains after Valentine's day at a discount price of $2 per unit.

Considering this information, how many keychains should Sarah order in October to maximize profit?