[A-Q1] Farmer Bob runs a pig farm and currently has as many male pigs as female pigs. Farmer Sue decides to buy two pigs from Bob and sends her son to pick them up, who just randomly grabs two pigs and throws them in the truck. Sue is angry because she wanted him to make sure one was male and one was female but her son doesn't remember what he purchased. What is the probability that one is male and one is female? After a moment, the boy recalls that at least one was a sow (female). Now what is the probability that the other pig is male? (You may suppose Farmer Bob has an infinite number of pigs ... that is the proportions do not change by purchasing a pig.) [A-Q2] Based on many years of experience, a teacher knows that there is a 75% chance that any given student in her class studies for an exam. Of those that study, 90% of them pass. Of those that do not study, only 20% of them pass. What is the probability that: a. A randomly selected student will pass the exam? b. A student will pass and not have studied? c. A student will not pass and have studied? [A-Q3] After examining Star Fleet historical files, Kirk realizes that a particular kind of attack on the Romulans that he is considering is very difficult, succeeding only 5% of the time. Because of the challenge, he asks Spock to build a simulation to predict whether an attack successfully achieves its objectives or not. Spock runs the simulation many times under different random historical conditions, which are then validated against historical real-world data. Spock's simulation is fairly accurate -- in only 3% of the cases does it indicate failure when there was an historical success and in only 4% of cases does it indicate success when there was an historical failure. Kirk plugs in the current situation he is considering into Spock's simulation and runs the simulation which predicts success against the Romulan scourge. But he can't cheat on this one, and so he is unwilling to commit the Enterprise to the battle unless the probability of an actual success is at least 70%. Should the Enterprise attack? Why or why not?