Question

A game has two doors, and you are asked to pick one. One door has twice as much money than the other. Using discrete math, consider the following: i. Your current door contains some unknown amount of money, called n dollars. ii. If you switch doors, you will double or lose half of the money. iii. The expected value of switching is expressed with: (1/2​×2n)+(1/2×2n​)=n+n/4=1.25n iv. Since this is higher than your current value of n, you should switch. You realize that if you did switch doors, the game could ask if you wanted to switch a second time - you would use the same logic and decide to switch again. You could end up switching forever, what was the flaw with the logic?

Answer 1

The flawed logic in the discrete math problem is that it fails to account for the fact that switching doors does not offer a continuous gain. The expected value of switching must consider the actual amounts behind the doors after each switch, which makes it static. Understanding the expected value, PDF table, and law of large numbers is crucial for calculating long-term outcomes in games of chance and informs decisions on whether to participate based on potential profit or loss.

Step-by-step explanation:

The flaw in the logic pertains to the misunderstanding of expected value when applied repeatedly in a situation with diminishing returns. In the discrete math problem described, we initially calculate the expected value of switching doors based on the assumption that one door has twice as much money as the other. However, in reality, if you constantly switch doors, you do not continuously gain;

instead, you alternate between doubling and halving the amount. Hence, after the initial switch, the expected value does not remain the same since the amounts behind the doors change; it becomes static and you're no longer guaranteed to keep increasing your money by just switching. The calculation (1/2×2n) + (1/2×n/2) simplifies to 1.25n, but applying this calculation repeatedly ignores the new baseline value after each switch.

The expected value, probability function, and law of large numbers all demonstrate we can calculate long-term outcomes of random events, but they do not guarantee specific short-term results and should be used considering the context of each situation. For instance, gambling games like betting on a biased coin or deciding whether to play a card and coin game involve understanding the expected value to determine if the game is profitable over many plays. However, the expected value can be misleading without considering the variable payouts and probabilities.

It's important to set up a PDF table to calculate expected profits for games of chance such as matching numbers or suit guessing. Understanding these concepts helps in making informed decisions about games that involve risk, and whether or not they should be played, based on their expected profit or loss.