You are a contestant on a game show and have won a shot at the grand prize. Before you are three doors. $1,000,000 in cash has randomly been placed behind one door. Behind the other two doors are the consolation prizes of dishwasher detergent. The game show host asks you to select a door, and you randomly pick one. However, before revealing the prize behind your door, the game show host reveals one of the other doors that contains a consolation prize. At this point, the game show host asks if you would like to stick with your original choice or to switch to the remaining door.

Write a function to simulate the game show problem. Your function should randomly select locations for the prizes, select a door at random chosen by the contestant, and then determine whether the contestant would win or lose by sticking with the original choice or switching to the remaining door. You may wish to create additional functions invoked by this function.

Next, modify your program so that it simulates playing 10,000 games. Count the number of times the contestant wins when switching versus staying. If you are the contestant, what choice should you make to optimize your chances of winning the cash, or does it not matter?