Based on some recipes I received from students, I decided to start making dumplings this spring.I decided I would pan fry these dumplings in a very large frying pan.The pan can fit up to 1000 dumplings at a time (it is VERY large).I cook them for ten minutes - ideally at least 3 minutes per side.An exhaustive study on the topic shows that dumpling lovers will be happy with them if 90% or more are cooked on both sides.

To simplify the process (and because I am lazy, yet oddly willing to spend weeks developing new and otherwise useless technology!) I developed a machine that shakes the pan for an instant at the 3 minute mark and then again at the 6 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 I cook 12 dumplings for a dumpling lover, what is the probability that that person would be happy with them?

b.Since I live in fear of upsetting a dumpling lover, I decided to cook 13 dumplings and serve the best 12.(I'll eat the other one myself, 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?