Dr Harris has recently received the following gifts: one partridge, two turtle doves, and three French hens. Unfortunately, she does not have space for all six birds so she selects two at random to keep. (a) What is the probability that she selects two of the same type of bird to keep? [6 marks] (b) Let R be the random variable counting the number of partridges kept, and I be the random variable counting the number of turtle doves kept. (i) Determine the joint probability mass function of the random variables R and T, and write it in the form of a table with six entries. [4 marks] (ii) Explain briefly how you could check your answer to (i). [2 marks] (iii) State the name for the distribution of R, and specify any parameters. [2 marks] (c) The two birds which are kept are to be fed on pears, from a tree growing by Dr Harris' blackboard. She observes that the number of pears produced each week is a Poisson random variable with the probability of zero pears being 0.1. If a partridge needs to eat two pears per week, while the other birds each need one pear per week, what is the probability that there is not enough fruit in a given week? Explain your method carefully and give your final answer to two decimal places. [6 marks]