STAT600 Probability

Question One  A coin is tossed until either a total of three heads or two tails have occurred.
(a) Write down the sample space for this experiment and find the probability of each event. (b) Suppose A is the event that the first coin is a head and B is the event that the last coin tossed was a head. Find the conditional probability of A given B. (c) Let X be the number of tosses up to and including the toss on which the last head occurs. Find the probability mass function of X. Take X = 0 if no head occurs. (d) Find E[X] in (c). What does its value mean? (e) Graph the distribution function F(x) in (c).
Question Two (10) To preserve anonymity in a survey of student drug taking habits the following procedure was devised. Each student was given a box which contained red and green beads in the ratio 2:1. When the box is shaken, one bead appears at random at a window on the box. Each student being surveyed enters a cubicle where the student shakes the box and observes the colour of the bead appearing at the window. The student agrees to answer truthfully either "yes" or "no" to one of the following questions depending on the colour of the bead observed: Red bead observed: Is your birthday in the first half of the year? Green bead observed: Have you ever smoked pot?
(a) Given 156 answered "yes" out of a random sample of 400 students, estimate the percentage of pot-smokers in the student population. (b) What assumption have you made in your answer?
Question Three)
Consider Example 1.30 on page 26 in the manual. Find the probability that two people in our class of 29 have the same birthday.
Question FourConsider the Titanic survival data. Are the events "being a female passenger" and "surviving" independent? Justify your answer with a calculation.