Question 1 [12 marks]
A list of all possible values of the discrete random variable X, together with their associated probabilities, is called a probability distribution. It is often helpful to show the probability distribution in a table. Stephanie is very forgetful. Every time she logs in to her online bank she only has a 40% chance of remembering her password correctly. She is allowed 3 unsuccessful attempts on any one day and then the bank will not let her try again until the next day.
(a) Draw a fully labelled tree diagram to illustrate this situation. [3]
(b) Let X be the number of unsuccessful attempts Stephanie makes on any day that she tries to log in to her bank. Construct a table to show the probability distribution of X. [4]
(c) Calculate the expected number of unsuccessful attempts made by Stephanie on any day that she tries to log in. [2]
(d) Calculate variance.of X. [3]
Question 2 [18 marks] Eight friends take a picnic to a cricket match. As her contribution to the picnic, Manisha buys eight sandwiches at a supermarket. She selects the sandwiches at random from those on display. The probability that a sandwich is suitable for vegetarians is independently 0.3 for each sandwich. Consider a Binomial distribution.
(a) Find the probability that, of the eight sandwiches, the number suitable for vegetarians is:
(i) 2 or fewer ; [2]
(ii) exactly 2 ;[1]
(iii) more than 3. [2]
(b) The weights, in kilograms, of men and women have the distributions N(78, 72 ) and N(66, 52 ) respectively.
(i) State two theorems of Expectation and two theorems of Variance. [4]
(ii) The maximum load that a certain cable car can carry safely is 1200 kg. If 9 randomly chosen men and 7 randomly chosen women enter the cable car, find the probability that the cable car can operate safely. [5]
(iii) Find the probability that a randomly chosen woman weighs more than a randomly chosen man. [4]
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