Monty Hall, the host of the TV game show "Let’s Make A Deal", shows you three curtains. One curtain conceals a valuable prize, while the other two conceal junk. All three curtains are equally likely to conceal the prize. He offers you the following "deal": pick a curtain, and you can have whatever is behind it. When you pick a curtain, instead of giving you what lies behind the curtain, Monty (who knows where the prize is) opens one of the remaining curtains to show you that there is junk behind it, and offers the following "new, improved deal": you can either stick with your original choice, or switch to the remaining (unopened) curtain. Amidst the deafening roars of "Stand pat!" and "Switch!" from the crowd, Monty points out that "previously your chances of winning were 1/3, and now, since you know that the prize is behind one of the two unopened curtains, your chances have increased to 1/2, and thus the new improved deal is indeed better."

Explanation:

I consider the Monty Hall game show problem to be a statistical illusion. This statistical illusion happens because your brain’s process of evaluating probabilities in the Monty Hall game show problem is usually on a false assumption. Statistical illusion is similar to optical illusions, the illusion seems more real than the actual answer.

To thoroughly view this statistical illusion, one needs to carefully break down the Monty Hall game show problem and identify the wrong assumptions. This process highlights how important it is to check that you’re clicking with the assumptions of a statistical analysis before trusting any of the results.

Monty Hall asks you to choose one of three doors in the show. One of the doors shields a prize and the other two doors contains no prize. You then loudly state out which door you pick, but you don’t get to open it right away. Monty then, opens one of the other two doors, and there, sit no prize behind it. At this stage, there are two closed doors containing one of which you picked. The prize sits behind one of the closed doors, but you can't tell which one.

Monty then asks you the ultimate question, “Do you want to switch doors?”. Majority of people assume that both doors are equally likely to have the prize. It seems like the door you chose has a 50/50 chance. Typically, there is no actual reason to change and most stick with their initial choice.  Theories were then speculated that switching doors doubles one's chances of winning the prize if they switch doors. Before, it was mere speculated theory but now, people are starting to see the sense behind it. Through a probability test, it is claimed that one has 66% chances of winning the prize if they switch doors over a 33% chance if they don't switch doors.