2. Your friend has three different tetrahedra (i.e., four-sided dice). You have the following information: . Each tetrahedron has each of its four surfaces labelled with 0 or 1. • Each tetrahedron has at least one of its sides labelled with 0, and one labelled with 1. A random experiment can be conducted with such a tetrahedron by throwing it and reading the label on the surface on which it comes to rest. a) For the tetrahedron that has the greatest number of surfaces labelled with 0 give a probability space that describes the corresponding random process. (2 marks) b) You create a binary number by rolling the same tetrahedron three times and writing down the number from left to right. For each possible tetrahedron what is the probability that the number obtained in this way is at least 2 when translated to base 10? (3 marks) c) For the situation from part b) create random variables by turning the binary numbers into the corresponding numbers in base 10. Calculate the expected value for the random variable X you obtain for the tetrahedron from part a). (2 marks) d) Given the situation from part c) calculate the conditional probability distribution given that the first throw shows 0. (3 marks) e) Your friend picks one of the three tetrahedra, and you are trying to work out which one it is. You do so by Bayesian updating. Your friend throws the tetrahedron for you and shows you the number labelling the bottom face, keeping the rest of the tetrahedron hidden from you. They do this three times, obtaining the numbers 0, 0 and 1 in that order. Carry out the corresponding calculation. Which tetrahedron do you think it is? (10 marks)