Michael planned to start cooking dumplings this summer and decided he would panfry these dumplings in a large pan that can fit up to 1000 dumplings at a time. He cooks them for 10 minutes (atleast 3 minutes per side). As per study, he knows that dumpling lovers will be happy with them if 90% or more are cooked on both sides.

In order to simplify the process, he developed a machine that shakes the pan for an instant at the three minute mark and then again at the six minute mark.The instant the pan shakes there is a 90% chance that any particular dumpling will flip over to allow the other side to cook.

a.If Michael cooks 12 dumplings, what is the probability that that dumpling lover would be happy with them?

b.Since he lives in fear of upsetting a dumpling lover, he decided to cook 13 dumplings and serve the best 12. He will eat the other one himself, without anyone noticing. What is the probability the dumpling lover would be happy then?

c.What is the 95% approximate confidence interval on the number of properly cooked dumplings for a full pan?