Rougier is planning a get together for his college classmates. He has contracted with a restaurant to provide food at this event. He has to pay $60/guest and the final number he commits to will decide how much it's going to cost. If he commits to say 50 people he has to pay $60*50=$3000 even if, say, 47 guests show up. The contract also calls for higher rate of $80 for any guest over the number he commits. So, if he commits to 50 and 53 show up, he has to pay $60*50+2*80=$3160. Rougier knows from planning these events that he can't exactly predict how many people will show up. Being a statistician, he decided to "Math" his way to an optimal solution. If he assumes that the number of people who are going to show up is normally distributed with mean of 60 and standard deviation of 3:

1. How many guests should Rougier commit to with the restaurant? 2. Suppose Rougier commits to 64 guests. What is his expected bill? 3. Suppose that the restaurant is willing to alter the contract so that if fewer than the number of guests they commit to show up, Rougier will get a partial refund. In particular, he only has to pay $25 for each "no-show." For example, if he commits to 50 but only 48 show up, he will have to pay 48 $60 +2 $25 = $2930. Now how many guests should he commit to? 4. The restaurant offers Rougier another option. He could pay $70 per guest, no matter how many guests show up; that is, he wouldn't have to commit to any number before the wedding. Should Rougier prefer this option or the original option ($60 per committed guest and $80 each guest beyond the commitment)?