A famous problem exists in probability, known as the Monty Hall problem. The Monty Hall problem is based upon a US game

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A famous problem exists in probability, known as the Monty Hall
problem. The Monty Hall problem is based upon a US game show called
“Let’s Make a Deal”. This question is based on a variation of the
Monty Hall problem. Suppose that you are on a television game show.
In the final round, you are given the choice between four boxes. •
Inside one of the boxes is a prize of £10,000. • Inside one of the
boxes is a prize of £5,000 • The other two boxes are empty, and
would lead to a prize of £0. The gameshow host knows the contents
of each of the boxes, but she gives you no indication of which box
contains the £10,000. Every time the game is played, the host
follows exactly the same procedure. You randomly select one of the
boxes. After you have made your choice, the host opens one of the
empty boxes. You are then offered the option to either stick with
your original choice, or to switch to a different box. You have
read that the optimal strategy in the Monty Hall problem is always
to switch to an alternative box, so you randomly choose between the
two other boxes. • For example, suppose that the boxes have labels
1,2,3,4. Initially you chose box 1, and the game show host shows
you that box 2 is empty. You then randomly choose between box 3 and
box 4. (a) Draw out a game-tree, illustrating this game. Carefully
define the two random variables you are considering here. (b) Write
down the marginal probabilities of initially choosing the following
prizes: i. £0 ii. £5,000 iii. £10,000 (c) Construct the conditional
probability distribution of the prize that you win at the end of
the game. i.e. Calculate the probability of winning £0, £5,000 or
£10,000 conditional on your initial choice having been a box
containing £0, £5,000, or £10,000. (d) Hence, calculate the
expected winnings of the game, conditional on initially have chosen
the following prizes i. £0 ii. £5,000 iii. £10,000 (e) Hence, use
the law of iterated expectations to calculate the expected winnings
from the game. (f) Suppose that you follow this strategy, and win
£5,000. What is the probability that you initially chose the
£10,000 box?