Note   : Whenever you use a formula, please detail in sentences exactly how/why it works for your particular solution (merely plugging values into a formula does not by itself demonstrate understanding). Proving a more general case of the formula(e) is another valid strategy.

QUESTION : A fair coin is tossed. If it lands heads, a fair four-sided die is thrown (with values 2,3,4,5). If it lands tails, a fair six-sided die is thrown (with values 3,4,5,6,7,8). Regardless of which die is used, Alice eats n grains of rice, where n is the largest prime factor of the die result (for example, the largest prime factor of 9 is 3).         

PLEASE SOLVE BOTH PARTS FOR THE ABOVE QUESTION

(a)  What is the conditional probability that the coin lands heads, given that Alice eats two grains of rice? (Please use an appropriate notation and/or justification in words, for each value that you give as part of your solution.) [8 marks]

 (b) Suppose that the entire experiment is conducted twice on the following day (starting with a new coin toss on the second run-through). What is the conditional probability that the coin lands heads on both run-throughs, given that Alice eats a total of seven grains of rice during the two run-throughs?                                                     (Do not count the two grains from part (a) in part (b); we assume two brand new experiments, each with a new coin toss. Start your solution by defining a suitable partition of the sample space. Please use an appropriate notation and/or justification in words, for each value that you give as part of your solution.)