Bayes’ Rule and Monty Hall Problem
In class, we learned about the Bayes’ Rule and how it plays a role in decision-making. It is a mathematical model used to measure the probability of one event occurring based on prior conditions related to that event. Bayes’ Rule allows us to update our beliefs based on the arrival of new information. In the herding experiment example, we are trying to predict the behaviors of the students after they draw a ball from the urn containing blue and red balls. Based on the ball they drew (red or blue), the students have to guess whether the urn they drew the ball from is majority red (MR) or majority blue (MB). Each student will have information about the ball he/she drew, which we refer to as the private signal. The private signal would update the prior (the probability of MR/MB), which is the probability of an event of interest without any additional prior information. Bayes’ Rule allows us to calculate the probability of the urn being MR or MB based on the private signal received, so we can reasonably predict the behavior of the students. Eventually, we see that it leads to the development of an information cascade when the public signal overcomes the private signal. At that point, the student would apply Bayes’ Rule and realize that it would be better for him to follow the public signal. Everyone after him would also follow.
Another interesting application of Bayes’ Rule is the Monty Hall Problem. It is based on a TV game show called Let’s Make a Deal and the puzzle is named after the host Monty Hall. In the game, there are three doors and the host hides a car in one of the three doors and a goat behind each of the other two. The contestant would pick one of the doors as his guess, then Mr. Hall would open one of the other two doors to reveal a goat. Now, the host gives the contestant the option to keep the choice to keep the door he has or switch to the other unopened door. What should the contestant do? Should he stick with his initial choice or switch? Or does it matter? At a first glance, one might say that there should be no difference in switching, because there’s a 50/50 chance for the contestant to win the car. However, Bayes’ Rule reveals that the odds for winning the car is 1/3 if the contestant sticks with his initial choice and the odds of winning increase to 2/3 if he switches.
The Bayes’ Rule plays a role in the Monty Hall Problem, because you gain more information about the closed door after Mr. Hall revealed one of the false doors. We can try to solve this problem through Bayes Rule. The three doors are Door A, Door B, and Door C. The odds of the car being behind one the three doors are all equal to 1/3 ( Pr(Car behind A) = Pr(Car behind B) = Pr(car behind C) = 1/3 ). Let’s say that the contestant picked Door B and the host revealed the goat behind Door C. Now, we can calculate the probability of the car behind Door B, given that the host opens Door C. The calculation is shown below:
Pr(Car behind Door B | Open Door C) =(Pr(Open Door C | Car Behind Door B)*Pr(Car Behind Door B))/Pr(Open Door C)
Pr(Open Door C) = Pr(Open Door C | Car Behind Door A)*Pr(Car Behind Door A)
+Pr(Open Door C | Car Behind Door B)*Pr(Car Behind Door B)
+Pr(Open Door C | Car Behind Door C)*Pr(Car Behind Door C)
Pr(Open Door C)=(1*1/3)+(1/2*1/3)+(0*1/3)=1/2
- Pr(Open Door C | Car Behind Door A) is 1, because the host only opens the door that has a goat.
- Pr(Open Door C | Car Behind Door B) is 1/2, because the host can pick either Door A or Door C.
- Pr(Open Door C | Car Behind Door C) is 0, because the host won’t open the door with the car.
Pr(Car behind Door B | Open Door C) =(1/2*1/3)/(1/2)=1/3
We can see that the probability of winning the car is just 1/3 if the contestant decides to not switch. The probability can be broken down based on the contestant’s two options (not switch and switch): 1/3 + x = 1 (the probability sums to one), in which x represents the probability of winning the car if he switches. This means that the probability of winning the car is 2/3 if the contestant switches. Thus, the contestant should switch, as his chance of winning the car doubles.
Sources:
https://www.nytimes.com/2014/09/30/science/the-odds-continually-updated.html?
https://www.norwegiancreations.com/2018/10/bayes-rule-and-the-monty-hall-problem/