KindiakMath

This post pertains the Monty Hall problem.

Problem 1. In a game show, there are three identical doors. Before the game starts, the host hides a prize behind one of the doors. He asks you to choose a door—you keep the prize (or lack thereof) from whichever door you choose. Whichever door you choose, he reveals one of the two remaining empty doors. He then asks if you would like to change your choice to the other closed door. Would you change?

Solution. Given doors 1, 2, 3, suppose without loss of generality that door 1 contains the prize. Without changing, the probability of winning the prize is 1/3. With changing:

• the initial choice 1 leads to the final choice being either 2 or 3 (whichever door the host does not open),
• the initial choice 2 leads to the final choice being 1,
• the initial choice 3 leads to the final choice being 1.

Thus, two out of three of the initial choices lead to the prize. Therefore, with changing, the probability of winning the prize is 2/3.

—Joel Kindiak, 5 Apr 26, 1906H