Question 1: You will receive a prize if both afair coin lands \"heads\" AND a fair die lands \"6\". After the coin isflipped and the die is rolled you ask if AT LEAST ONE of theseevents has occurred and you are told \"yes.\"

a) Use an event tree to help calculate the probability ofwinning the prize

b) Formally calculate the probability of you winning the prize,whilst answering these questions in each step of your answer

i. Specify the joint distribution, ?(?,?,?,?), in terms of itsconstituent conditional distributions

ii. Specify the full prior probabilities for the coin, ?(?) andthe dice, ?(?), events

iii. Specify the full conditional distribution for the eventthat the coin is heads or dice is six, ?=?∪?

iv. Specify the full conditional distribution for the event thatthe coin is heads and dice is six, ?=?∩?

v. Use the fundamental rule to derive the distribution for thecoin and dice events given the event that the coin is heads or diceis six, ?(?,?|?=????)

vi. Calculate the probability of observing that the coin isheads or dice is six, ?(?=????)

vii. Specify and calculate the posterior distribution for thejoint probability of the coin and dice events given theevent that the coin is heads or dice is six, ?(?,?|?=????)

viii. Derive the marginal distribution for the event that coinis heads and dice is six given we know the event heads or six,?=???? | ?=???? , has occurred

ix. Calculate the marginal probability that the coin is headsand dice is six given we know the event heads or six,?(?=????|?=????)

x. Calculate the probability of you winning the prize

Note: Your formal calculation must includemathematical notation and the derivation of every step in thecalculation without ambiguity. Your model must include thesevariables: ? for the coin, ? for the dice, ? for the event thatcoin is heads or dice is six and ? for the event that coin is headsand dice is six. Correct answer that take short cuts or ignore thefull set of variables will be penalised. Note, also by ‘full’distributions above it is meant that all relevant states for thevariables are used in the calculations.