1 We have the following sample Space 8 : {a,b,c,d,e} P is a probability function as follow: P({a}) = P({b})a P({c=}) = P({d}) = P({}) = 2P({a}) if A : {05, b} and B : {a1 c} and C : {3, b1 0}, find the following probabilii ties. i) P(A) and P(B) and P(C) ii) P(A|B) iii) P(A|C') 2 For an}r events A and B with 10(3) > U, Show that P(A|B) + P(A"|B) : 1 (Hint: use the conditional probability formula, distributive law and axiom 3 of probability) 3 Seventy percent of the light aircraft that disappear while in ight in a cer tain country are subsequently discovered. Of the aircraft that are discovered, 50% have an emergency locator, whereas FREE: of the aircraft not discovered do not have such a locator. Suppose a light aircraft has disappeared. a} If it has an emergency locator, What is the probability that it will not be discoverle? b} If it does not have an emergency locator, what is the probability that it will be discovered? 4- Airlines sometimes overbook flights. Suppose that for a plane with 50 seats, 55 passengers have tickets. Define the random variable Y as the num- ber of ticketed passengers who actually show up for the flight. The probability mass function of Y appears in the accompanying table. y 45 46 47 48 49 50 51 52 53 54 55 p(y) 0.05 0.1 0.12 0.14 0.25 0.17 0.06 0.05 0.03 0.02 0.01 a) What is the probability that the flight will accommodate all ticketed passengers who show up? b) What is the probability that not all ticketed passengers who show up can be accommodated? c) If you are the first person on the standby list (which means you will be the first one to get on the plane if there are any seats available after all ticketed passengers have been accommodated), what is the probability that you will be able to take the flight? What is this probability if you are the third person on the standby list? 5- An insurance company offers its policyholders a number of different pre- mium payment options. For a randomly selected policyholder, let X = the number of month between successive payments. The cdf of X is as follows: 0 r