# Homework #2

Due: Thursday, February 8, 2018 at 11:59 p.m.
Points: 100

The Monty Hall problem is a very famous problem in probability. It’s based on an old TV show called “Let’s Make a Deal”. The stage of that show had 3 doors numbered “1”, “2”, and “3”. Behind one of the doors was a valuable prize (like a new car); the other two contained gag gifts (like a goat or a can of cat food). The host, Monty Hall (hence the name of the problem), would choose someone from the audience and ask them to pick a door. The contestant would choose one, say door “2”. Monty would then open one of the other doors that always had a gag gift behind it (say, door “1” for our example). He would then ask the contestant if he or she wanted to stay with door “2”, or change their selection to door “3”. The problem is to determine which action – keep or change – gives the contestant the greater probability of selecting the door with the real prize.

We’re going to answer this question by simulation — the technique is called a Monte Carlo method. Basically, we play a large number of games on the computer, always switching doors (or never switching doors), and record whether we won. We then divide the number of times we won by the number of trials, giving a number between 0 and 1 (inclusive). This is the probability that the strategy will cause the contestant to win.

We’re going to build this program in steps, because that will simplify writing it.

1. (10 points) The basis for this simulation is a function that generates numbers randomly. Write a function that uses the Python random number generator (see section 4.4) to generate one of the numbers 1, 2, and 3 randomly. Then write a small program that calls this function 50 times and prints each number generated. Print the numbers on the same line with no intervening spaces.

Input. The program and function take no input.

Output. A list of 50 numbers (1, 2, or 3) on the same line.

2. (50 points) Next, we will write functions to simulate playing one game. The basic approach is to generate a random number representing the door behind which the real prize sits, and another random number representing the door that the contestant initially selects. Monty opens the remaining door. Then, have the contestant switch doors (or not switch doors), and see if the contestant winds up with the door behind which the prize sits.

Write two different functions to do this. The first, called montyalways(), has the contestant always changing doors after Monty opens the third door. The second, called montynever(), has the contestant never changing doors after Monty opens the third door. Both functions should return the Boolean value True if the contestant wins, and the Boolean False if she does not. Use the function you wrote in part 1 to generate the door number.

Input. The functions take no input.

Output. The functions return True or False depending on whether the contestant wins or does not win. They do not print anything.

Hint: Before you start programming, think about the best approach for these functions. There is a very simple way to do them.

3. (40 points) Now we will simulate a large number of games. Write a program that asks the user for the number of games to be played. Then play one set of games for the contestant always changing the door and another set for the contestant never changing the door. Print the resulting (decimal) fraction of times that the contestant wins, and the number of games won.

Input. The number of games to be played. This must be a positive integer. Remember to handle invalid inputs gracefully, by printing an error message and exiting the program.

Here is what a correct input should look like: (the red text is what you type):

Number of games to play: 100000
If the input is invalid:
Number of games to play: hello
and then the program exits.

Output. The output of your program must look like this:

Out of 100000 games:
Always switching wins: 0.6680600 (66806 games)
Never switching wins: 0.3346300 (33463 games)
Important note: your numbers may be different.