MA10211. Statistics and Probability I. Example Sheet Five Hand in ...

MA10211. Statistics and Probability I.Example Sheet FiveHand in your solutions by noon, Wednesday week 7 or earlier if yoututor so requests.1. Let X denote the result of a roll of a biased die. If X takes values1, 2, 3, 4, 5, 6 with probabilities 1/4, 1/4, 1/8, 1/8, 1/16 and p, respectively.Find (i) the value of p; (ii) P(X ≤ 3) (iii) P(X = 6 |X even).2. Suppose a die with faces numbered 1, 2, 3, 4, 5, 6 is biased so the probabilityof a given number turning up is proportional to that number.Determine the probability mass function, p(x), for the score, X, on aroll of the die. Let E be the event that X is an even number. WriteE in terms of events of the form {X = i}. Calculate (i) P(E); (ii)P(X = 5|E c ); (iii) P(E|X < 5).3. A test has ten questions which can be answered as true or false. Supposea certain student decides to guess each answer. Let X be thenumber of questions the student answers correctly. Write down thedistribution for the discrete random variable X. If the pass mark is8, what is the probability the student passes? Find the conditionaldistribution of X given the student passes, P(X = r |X ≥ 8).4. At each play of a game, two jokers and the ace of hearts are shuffledtogether and the player tries to guess which card is the ace. (a) Give thedistribution for the number of attempts, Z, required until the first timethe ace found. Calculate P(Z > 2). By suitable geometric summations,find (i) P(Z ≤ r) where r ∈ {1, 2, . . . }. Can you find a quick way tocalculate this probability? (ii) the probability that Z is even.(b) What’s the probability that more than 2 aces are found after 9games are played?5. Let X denote the number of road accidents in one hour in a certaintown and suppose that X is Poisson distributed with parameter 1.5.Write down the distribution for X, that is, the probabilities P(X = r)for r = 0, 1, 2, . . . . Evaluate (i) P(X = 0); (ii) P(X > 3).6. ‡ A fair coin is tossed 2n times. Find the probability of obtaining anequal number of heads and tails, and show that this probability is adecreasing function of n.1

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