Monty Hall Problem — Theoretical Vs Experimental Probabilities

Let’s Make a Deal (1973)

The Monty Hall problem is a famous, seemingly paradoxical problem in conditional probability and reasoning using Bayes theorem. The main purpose of this post is to understand Monty Hall Problem and the difference between experimental probability and theoretical probability.

What is Monty Hall problem?

The Monty Hall Problem gets its name from the TV game show, Let’s Make A Deal, hosted by Monty Hall. The scenario is such: you are given the opportunity to select one closed door of three, behind one of which there is a prize. The other two doors hide “goats” (or some other such “non-prize”), or nothing at all. Once you have made your selection, Monty Hall will open one of the remaining doors, revealing that it does not contain the prize. He then asks you if you would like to switch your selection to the other unopened door, or stay with your original choice.

Monty Hall problem

From the above problem statement it is understood that there is a possibility that we can win this car and probability will help us to increase this odd. So let’s see how probability could help in this case. Believe me, by the end of this post you will understand how probabilities could change our lives. Before jumping into the problem, let’s understand two types of probabilities.

• Theoretical probability
• Experimental probability

bag of marbles

Theoretical probability

Consider a bag of marbles. It consists of two different colour marbles- red and blue. Let’s say there are 50 red marbles and 50 blue marbles. Theoretically, the probability of picking a red marble can be calculated as:

p( picking red marble ) = no. of red marbles / total number of marbles in the bag = 50/100

Hence theory states that there is a 50% chance that the marble what we pick from the bag is of red color.

theoretically: p( picking red marble ) = 50%

Experimental probability

Now let’s say we start doing some experiments. So we literally take a bag with 50 red marbles and 50 blue marbles and then start picking the marbles. Every time we pick a marble, we will note down the color of the marble. After 10 experiments, our result might seem to have 7 red marbles and 3 blue marbles over 10 experiments. Contradiction to the previous theoretical probability( which is 50% ) through experiments we observed:

p( picking red marble ) = no. of experiments in which red marble observed / total number of experiments = 7/10

Hence experiment stated that there is 70% chance that the marble what we pick from the bag is of red color.

experimentally: p( picking red marble ) = 70%

What we observed? Even though theoretical probability clearly defined the outcome, it failed over series of experiments. Hence experimental probability is considered to be more significant over the theoretical probability. Let’s understand practical implications.

Monty Hall Problem

Theoretical probability

Suppose you’re on a game show, and you’re given the choice of three doors: Behind one door is a car; behind the others, goats. So what is the probability of finding the right door?

Theoretically: p( picking the right door) = 1/3 = 33.33%

You pick a door, say No.1, and the host, who knows what’s behind the doors, opens another door, say No.3, which has a goat. He then says to you, “Do you want to pick door No.2?" Now what is the probability of finding the right door?

Theoretically: p( picking the right door | one of the FALSE door has been revealed) = 1/2 = 50%

Because one of the door which had “goat” has been revealed. Now there are only 2 doors left — the one which you have selected and the other one is still a mystery. Hence theoretical probability helped us to make some decision. Let’s see how experimental probability would help us.

Experimental probability

There are two ways to do this experiment. Let’s list down what are the possible experiments.

Experiment 1: You will change your choice once the host reveals the false door.

Experiment 2: You will not change your choice even after the host reveals the false door.

Let’s list down all the possible ways of performing experiment 1. There are 3 possibilities those are:

• You pick the first door and the host will reveal the second door.
• You will pick the second door and host will reveal the first door.
• You will pick the third door and host will reveal either first door or second door.

Notice that in this experiment we are not changing the door once we select it.

P- Your pick. R-revealed by host

Experimentally: p( picking the right door ) = 1/3 = 33.33%

Now, Let’s list down all the possible ways of performing experiment 2. There are 3 possibilities which are:

• You pick the first door, the host will reveal the second door and you will change to third door.
• You will pick the second door, the host will reveal the first door and you will change to third door.
• You will pick the third door, the host will reveal either the first door or the second door and you will change to either of the first or second door.

Notice here we are changing our decision once the host reveals the one of the falls door.

P- Your pick. R-revealed by host

Experimentally: p( picking the right door ) = 2/3 = 66.66%

Conclusion:

Theoretical probability stated that

• There is a 33.33% chance of picking the correct door in the first attempt and a 50% chance of picking the correct door once the host reveals one of the false doors.

Experimental probability stated that

• There is 33.33% chance of ending up with the correct door if you won’t change your decision once the host reveals one of the false door and 66.66% chance of ending up with right door if you change your decision once the host reveals one of the false doors.

Hence experimental probability increased our odds of winning the game.