Question

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

a) Use an event tree to help calculate the probability of winning 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 its constituent conditional distributions

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

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

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

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

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

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

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

ix. Calculate the marginal probability that the coin is heads and 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 include mathematical notation and the derivation of every step in the calculation without ambiguity. Your model must include these variables: ? for the coin, ? for the dice, ? for the event that coin is heads or dice is six and ? for the event that coin is heads and dice is six. Correct answer that take short cuts or ignore the full set of variables will be penalised. Note, also by ‘full’ distributions above it is meant that all relevant states for the variables are used in the calculations.

Answer 1