Consider the UK national lottery, where there are 49 possible numbers and six balls are selected wityhout replacement. Now consider Mrs Prime who always selects balls 2, 3, 5, 7, 11, 13. Let the random variable X be the correct number of choices made by Mrs Prime and the random variable T be the total of the six numbers drawn. For example, if the six balls drawn are 5, 10, 15, 20, 25 and 30, then the outcomes are X = 1 and T = 105.

i) Find E[T], E[T|X = 6] and E[T|X = 0].

ii) Are X and T independent random variables?

iii) Find E[X2 ] and Var[X].