Question

1. (a) Consider a modified version of the Monty Hall problem. In this version, there are 8 boxes, of which 1 box contains the prize and the other 7 boxes are empty. You select one box at first. Monty, who knows where the prize is, then opens 6 of the remaining 7 boxes, all of which are shown to be empty. If Monty has a choice of which boxes to open (i.e. if the prize is in the box you chose at first), he will choose at random which one of the boxes to leave unopened.

i. Suppose that you have chosen Box 1, and then Monty opens Boxes 3 to 8, leaving Box 2 unopened. After we have observed this, what is the probability that the prize is in Box 1, and what is the probability that it is in Box 2?

ii. How should a risk-neutral decision-maker use the probabilities computed in i. to inform their strategy?

(b) ‘If there is uncertainty about some monetary outcome and you are concerned about return and risk, then all you need to see are the mean and standard deviation. The entire distribution provides no extra useful information.’ Do you agree or disagree? Provide an example to back up your argument.

(c) It is assumed that inter-person arrival times at a bank during the peak period covering lunchtime follow an exponential distribution with a mean of 20 seconds. An Excel analysis was conducted as follows:

Times between arrivals at the bank: Mean time between arrivals: 20 seconds. Cell B5 = 0.0498

Explain what the value of 0.0498 in cell B5 represents and why this function is used.

Answer 1

Sol:

a).

Probability that the prize is in Box 1 = Probability that the prize is in Box 2 = 1/8

Given that you have chosen Box 1,

Probability that Monty opens Boxes 3 to 8 if the prize is in Box 1 = 1/7

Probability that Monty opens Boxes 3 to 8 if the prize is not in Box 1 = 1

Probability that the prize is in Box 1

= (1/7*1/8)/(1/7*1/8 + 1*1/8)

= 1/8 = 0.125

Probability that the prize is in Box 2

= 7/8 = 0.875

b).

c).

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