(a) only F occurs

P1. Let E, F, G be three events. Find expressions for the events that out of E, F, G,(a) only F occurs,(b) both E and F but not G occurs,(c) at least one event occurs,(d) at least two events occur,(e) all three events occur,(f) none occurs,(g) at most one occurs,(h) at most two occur.(Hint: The answer for (c) is E ∪ F ∪ G.)P2. Let E, F, G be three events. Find P{E ∪ F ∪ G} function of P{E}, P{F}, P{G},P{E ∩ F}, P{E ∩ G}, P{F ∩ G}, and P{E ∩ F ∩ G}.P3. A fair coin is to be tossed until a head appears twice in a row. What is the samplespace for this experiment? What is the probability that the coin will be tossed exactlyfour times?P4. Suppose that two fair dice are rolled.(a) What is the probability that both rolls show even numbers?(b) What is the probability that the sum of the two numbers is 10?(c) What is the probability that the two numbers differ by at least 3?P5. Consider a group of 10 persons. What is the probability that no two persons of thegroup have the same birthday?P.6 In a gambling casino game you select a number between 1 and 6 before three fairdice are rolled. The casino pays you as many dollars as the number of dice that matchyour selection. If there is no match, you pay the casino $1. What is the expected pay-offof this game?P.7 Suppose that an airplane engine will fail, when in flight, with probability 1 − pindependently from engine to engine. Suppose that the airplane will make a successfulflight if at least 50% of its engines remain operative. For what value of p is a two-engineplane preferable to a four-engine plane?

P.8 A post office is run by two clerks. Mr. Smith enters the office and finds the twoclerks busy serving Mr. Jones and Mr. Brown. The amount of time a clerk spends with acustomer is exponentially distributed with mean 20 minutes.(a) What is the probability that, out of the three customers Mr. Smith is the last to leavethe post office?(b) What is the probability that Mr. Smith waits for 30 minutes before being served?P.9 The weight of individuals who seek a helicopter ride in a park is 180 lb with astandard deviation of 5 lb. The helicopter can carry five persons but has a maximumweight capacity of 1000 lb. What is the probability that the helicopter cannot fly with fivepersons aboard?P.10 Suppose independent trials, each having a probability p of being a success, areperformed until k success occur. What is the probability that n trials are performed?