Statistics

    
Exercise 1: A fair six-sided die is tossed until a 6 is observed. Let X be the number of tosses until (and including) the first 6 is observed.
a) Find the probability that at least 5 rolls are required to obtain the first 6
b) Given that a 6 was not rolled on the first 3 rolls, find the probability that at least 5 more rolls are needed to obtain the first 6.
c) Your two answers should be equal. What property of the geometric random variable is illustrated in this exercise?

Exercise 2: Consider a student taking a multiple-choice test. On a given question, either the student knows the answer, in which case he answers it correctly, or he does not know the answer, in which case he guesses hoping to guess the correct answer. Assume there are five multiple-choice alternatives. Let p be the probability that the student
knows the correct answer. Let us assume that the probability that the student gets the correct answer given that he guesses is 1/5. Show that the probability that the student 
knowsthe correct answer given that the student gotthe correct answer is 5p/ 4p+1
HINTUse the Theorem of Total Probability together with Bayes’ Theorem. Carefully define the events.

Exercise 3: In this game, there are five fair six-sided die. Begin by rolling all five dice (round 1). The objective is to obtain five 6s in exactly two rounds. For example, if two 6s were rolled in the first round, then you roll the remaining three dice in the second round in the hopes of rolling three 6s. If only one 6 was rolled in the first round, you roll the remaining four dice in the second round in the hopes of rolling four 6s. If no 6s are rolled in the first round, then all five dice are rolled in the second round in the hopes of rolling five 6s. And if five 6s are rolled in the first round, the game is over! But you don’t win because it did not take you exactly two rounds to obtain the five 6s (and you are not allowed to try again). Use the Theorem of Total Probability to find the probability of rolling five 6s in exactly two rounds. You will make good use of the binomial probability distribution.