12 A multiple choice test has 20 questions, each with five possible options, only one of which is right. A student guesses the answer to each question, with an equal probability of guessing any answer correctly. a What is the probability that he gets fewer than five correct answers? b State a further assumption you made in your calculation in part a. 13 On a particular day, 20 babies are born at a certain hospital. Denote by G the number of girls. a State two assumptions you need to make so that G can be modelled by the distribution B(20, 0.5). b Under these assumptions, find the probability that more than 15 of the babies are girls.Optional extension questions 15 X-B (8, p) . If P (X - 5) - P(X - 6) find the value of p. 16 X is a binomial random variable, where the number of trials is 4 and the probability of success of each trial is p. Find the possible values of p if P (X - 2) - 0.3456. 17 Over a one-month period, Ava and Sven play a total of n games of tennis. The probability that Ava wins any game is 0.4. The result of each game played is independent of any other game played. Let X denote the number of games won by Ava over the one-month period. a Show that P (X - 2) can be given as P (X - 2) - 2n (n - 1) x 0.6" . 9 b If the probability that Ava wins two games is 0.121 correct to three decimal places, find the value of n.12 a 0.630 b His guesses are independent of each other 13 a Both genders are equally likely. Genders of different babies are independent of each other. b 0.00591 15 p = 16 p - 0.6 or 0.4 17 a Proof b n = 10