ID : 43

2. a} b} There are 10 red balls and 15 white balls in an urn. Suppose you pick balls randomly from the urn one at a time and without replacement until you get 5 red balls. Let the random variable X represent the number of balls you picked to get 5 red halls. Find the PMF ofX. Note: Balls are picked without replacement, probability of picking a ball of particular color changes after each pick up. Also, the experiment ends with the pick up of a red bail. Suppose you need to know whether a group of IE") peoples in your area have a certain disease or not. Say there is a special blood test available and you need to test the blood of ltltl peoples. Assmning that the test is expensive, you decide to divide the peoples into 1t] groups with 1D peoples in each group. Now blood samples of all the It] peoples of a group are pooled and analyzed together, instead of doing this for each individual separately. If the test result of a group is negative, a single test is enough for the group. However, if the test result is positive for a group, each of the It] people in the group is individually tested again. Suppose the probability that a person is infected by the disease is t]. IQ for all peoples independent from each other. Let X represent the number of groups with at least one people infected. Find E[X]. Suppose you have two coins one of which is a fair coin and the other one is a biased coin with probability of getting head is p, where p = lt[mod {your_student_id, 3) + 3], mod returns the remainder of the division of the rst number by the second. You pick one coin randomly and give the other coin to your friend. Then you and your friend continue to toss the respective coins repeatedly until both of you get the same outcome {an outcome of a toss is either a head or a tail). Let X be the number of tosses required. Find the PMF of X