If getting the Monty Hall puzzle (explained below) right, if understanding the odds involved in making the best possible decisions, was a requirement to hold public office, how many current political leaders would fail to qualify? Competence in understanding mathematical probabilities is arguably essential to making the best decisions in governance. Think about that. Here is the Monty Hall problem with the non-intuitive solution explained:
The Monty Hall problem is a probability puzzle based on a game show scenario. In this variation of the problem, after making your initial guess, the host, Monty Hall, always opens a losing door. Let’s walk through the solution step by step:
1. Initially, you are presented with three closed doors: Door 1, Door 2, and Door 3. Behind one of the doors, there is a valuable prize, while the other two doors hide goats, which are considered losing options.
2. You make your first guess by selecting one of the three doors. Let’s say you choose Door 1.
3. After your initial guess, the host, Monty Hall, who knows the location of the prize, opens one of the remaining doors that conceals a goat. Crucially, he will always choose a losing door.
4. At this point, there are two doors left: the one you initially chose (Door 1) and another unopened door (let’s say Door 2 since Monty opened Door 3).
5. Now comes the critical decision point. You have two choices: stick with your original guess (Door 1) or switch to the remaining unopened door (Door 2).
6. The solution is that it is always more advantageous to switch your choice to the other unopened door. Statistically, this increases your chances of winning the prize.
Why does switching increase your chances?
By switching, you effectively take advantage of the probability that Monty knows where the prize is located and will always open a losing door. To illustrate this, let’s analyze the probabilities:
– Initially, the probability of selecting the door with the prize is 1/3 (33.33%). The probability of selecting a losing door is 2/3 (66.67%).
When Monty reveals one of the remaining losing doors, the probabilities update:
– If your initial guess was correct (1/3 chance), switching would result in a loss (0% chance of winning the prize).
– However, if your initial guess was wrong (2/3 chance), then switching would lead to a win (100% chance of winning the prize).
By switching, you take advantage of the higher probability (2/3) of initially choosing a losing door.
Hence, switching results in a higher chance of winning compared to sticking with your initial guess (1/3 chance).
This solution to the Monty Hall problem, in the case where the host always opens a losing door after your first guess, demonstrates the statistical advantage of switching your choice to maximize your probability of winning the prize.
It is one thing to read this and think, “Oh, okay, that makes sense,” but still not really get it. It is another thing entirely to understand this problem and to understand it in this context. It is a third level to not only take this lesson to heart, but to get this lesson well enough to apply it to more practical real-world problems as appropriate.
Having enough information at your fingertips is as important as understanding probabilities. For example, what are the odds of getting a two headed goat behind one curtain, a goat that is genetically able to turn sugar into vitamin C? Would winning that prize and specifically selling a two headed vitamin C making goat to a circus or a research lab bring you more of a reward than a new car? Hint: This is a “hopium” puzzle.