Dorothee is planning her daughter Elsa's birthday party, which will take place at the Young Chefs Academy

$($YCA$)$ in Frisco. The kids will make pizza from scratch, cook it then eat it$.$ One week before the party date,

Dorothee has to communicate to the YCA a fixed number of kids and this number is used to determine the

cost of the party. Specifically, the price charged by YCA is equal to $$18$ times the quoted number of kids. If$,$

on the day of the party fewer kids show up than the quoted number, Dorothee does not get a refund. If$,$ on

the other hand, more kids show up than the quoted number, she has to pay $$25$ per extra kid. Separately,

Dorothee will also buy some "goodie bags" at Party City $($Elsa$\text{'}$s favorite store$)$ for the kids to take home with

them after the party. These will be purchased one week before the party and cost $$5$ per bag. If$,$ after the

party, there are some goodie bags left, Dorothee's friend Mary will buy them back from her for $$3$ each. If$,$

on the other hand, there are not enough goodie bags on the day of the party, Dorothee will run to the

nearest store and buy a few small gadgets and sweets at a cost of S$8$ per extra goodie bag needed. Elsa

has invited $20$ of her best friends. Based on this number, the possible number of kids coming to the party,

and the corresponding probability, is given in the following table:

What is the expected number of kids that will show up on the day of the party: klds

Suppose Dorothee tells the YCA that $17$ kids will come to the party and picks up $17$ goodie bags from

Party City but in fact $18$ kids show up on the day of the party. How much is her total cost? $

Suppose Dorothee tells the YCA that $17$ kids will come to the party and picks up $17$ goodie bags from

Party City but in fact only $14$ show up show up on the day of the party. How much is her total cost? $

Suppose Dorothee decides to tell the YCA that $17$ kids will come to the party and picks up $17$ goodie

bags from Party city.

What is her expected total cost? $

What is the expected number of goodie bags which will be sold to Mary after the party?

goodle bags

What is her probability of running out of goodie bags $($i$.$e$.,$ stock$-$out probability$)?$

$\%$

What proportion of goodie bags given to the kids can she expect will be coming from the ones

bought at Party City $($i$.$e$.,$ fill rate$)?\%$

What is the probability that her total cost for the party $($YCA plus goodie bags$)$ is above $$400?$

$\%$

Assume that the number of kids Dorothee quotes to the YCA is also the number of goodie bags she

buys at Party City one week before the party. What should this number be if Dorothee wishes to

minimize her expected total cost, that is$,$ the expected total cost of the party $($YCA plus goodie bags$)?$

klds

How many goodie bags should Dorothee buy at Party City to make sure there is less than a $10\%$

chance that she needs to run to the nearest store to buy gadgets for the missing goodie bags?

goodle bags

How much would Dorothee expect to spend on the party if she could see the future, that is$,$ if she

could predict for sure how many kids will come to the party? $

Calculate the expected mismatch cost if Dorothee communicates a number of $17$ kids coming to the

party and purchases $17$ goodie bags at Party City. $