Lesson 10

This is the so-called Monty Hall paradox (host of the show Let's Make a Deal). It is called a paradox, as is often the case, only because the solution contradicts the intuition of many people. At one time, it caused a heated discussion among both enthusiasts and mathematicians. The common reasoning is: there is no point in changing the box because the participant has not received any new information, as there is always a box without a prize that the host can open.

In reality, changing the choice is more beneficial! And there is a simple explanation for this. If you do not change the box, you win only if you initially guessed the box with the prize, which happens with a probability of \(1/3\). On the other hand, if you change your choice, you win if you initially guessed incorrectly, and the probability of this is \(2/3\).

Now let's draw a decision tree for the strategy where the player changes the box. When drawing the tree, we assume that the prize is in a random box, the player selects a random box, and the host selects a random empty box (if they have a choice).

Decision tree for the Monty Hall paradox. The first level shows the number of the box containing the prize. The next level shows the player's choice. The last level shows the box that the host opens.