1. To test the theory that students habitually travel on trains without tickets, a survey of...

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1. To test the theory that students habitually travel on trains without tickets, a survey of university students has been taken. The students in the sample were asked if they have ever travelled on a train without a ticket, followed by whether they plan to travel without a ticket in future. 40% of students admitted to having travelled without tickets in the past. Of those that have already travelled without tickets, 83% plan to travel without tickets in future. Of those that have never travelled without tickets, 43% plan to travel without tickets in future. (a) Derive the formula and calculate the probability that a student has already travelled without a ticket given that s/he plans to travel without a ticket in future. [30 Marks] (b) Derive the formula and calculate the probability that a student has never travelled without a ticket given that s/he does not plan to travel without a ticket in future. [30 Marks] (c) The combination of answers to the survey questions could be 'I have and I will', 'I have and I will not', 'I have not and I will' or 'I have not and I will not'. Describe how you would simulate the process of observing the combination of answers. [40 Marks] 1. To test the theory that students habitually travel on trains without tickets, a survey of university students has been taken. The students in the sample were asked if they have ever travelled on a train without a ticket, followed by whether they plan to travel without a ticket in future. 40% of students admitted to having travelled without tickets in the past. Of those that have already travelled without tickets, 83% plan to travel without tickets in future. Of those that have never travelled without tickets, 43% plan to travel without tickets in future. (a) Derive the formula and calculate the probability that a student has already travelled without a ticket given that s/he plans to travel without a ticket in future. [30 Marks] (b) Derive the formula and calculate the probability that a student has never travelled without a ticket given that s/he does not plan to travel without a ticket in future. [30 Marks] (c) The combination of answers to the survey questions could be 'I have and I will', 'I have and I will not', 'I have not and I will' or 'I have not and I will not'. Describe how you would simulate the process of observing the combination of answers. [40 Marks]