Question

The Monty Hall problem is named for its similarity to the Let's Make a Deal television game show hosted by Monty Hall. The problem is stated as follows. Assume that a room is equipped with three doors. Behind two are goats, and behind the third is a car. You are asked to pick a door, and will win whatever is behind it. Let's say you pick door 1. Before the door is opened, however, someone who knows wh at's behind the doors (Monty Hall) opens one of the other two doors, revealing a goat, and asks you if you wish to change your selection to the third door (i.e., the door which neither you picked nor he opened). The Monty Hall problem is deciding whether you change your selection or not that has a better chance of winning the car . It’s common sense that if not to change, the probability of winning is 1/3 but what about changing the selection.

Simulating this game using SAS, for each round, program the fol lowing, 1) Assigning two goats and a car to three doors randomly 2) Picking a door randomly 3) Picking one of the two remaining doors to open but must showing the goat 4) Changing the selection to the remaining door 5) Deciding the result Repeating these steps for 100 round s , generating a data set including the following five variables, the round number, the door the car is in, the door chosen initially, the door chosen after switching, and the result(win/lose). Showing the data set and r eporting the frequenc y of the i nitial door chosen, the frequenc y of the door chosen at the end, the average rate of winning.

Answer 1

One can select 1 door out of 3 in starting of the game so the probability of originally selecting the door with the new red Corvette behind is.

After you selected 1 door and hosts open the second door you can either switch or not switch. Let W represents winning the new red Corvette. Now there are two possibilities:

Case 1: If you make a selection and do not switch then the probability of winning is

Case 2: If you make a selection and switch then there are only 2 possibilities.

(a) You originally had the winning door and switched to a losing door.

Let A represents this event.

(b) You originally had the loosing door and switched to a winning door.

Let B represents this event.

You cannot switch from a losing door to another losing door. Monty Hall always opens a losing door. If you have a losing door and Monty Hall opens a losing door, then the only possible door to switch to is the winning door. Since you win if B occurs so the probability of B is equal to probability of W. Therefore

Since if do not switch then the probability of winning is less so the better strategy is to switch.

A simulation of n trails can be done as follows:

Step 1: Randomly select door 1, 2 or 3 to hold the prize (Corvette).

Step 2: Randomly select door 1, 2 or 3 as your original choice.

Step 3: Let m represents the number of times the doors match. Find out m.

Step 4: Estimated probability of winning if you do not switch is

It should be approximately.

Step 5: Estimated probability of winning if you switch is

It should be approximately.

The following Minitab commands produce the desired simulation, where column 1 indicates the winning door and column 2 indicates your choice.

MTB > RANDOM 100 C1 C2;

SUBC> INTEGER 1 3

MTB > LET C3= C1-C2

MTB > TABLE C3

A zero in C3 indicates a match and that the original choice would win the Corvette. The simulation produced for this manual yielded 35 0’s in C3. Since there are 100 trials so

The estimated probabilities

The estimated probabilities