Second Markdown Assignment

Cucio, Quintos, Tan, Villar, & Villareal

July 16, 2021

The Monty Hall Problem

The Monty Hall Problem is a famous paradoxical problem based on the television game show, Let’s Make a Deal. The problem is named after the original host, Monty Hall.

The game is simple: There are 3 doors to choose from and behind one of those doors is a grand prize of one luxurious car while the others have a goat. After the player chooses a door, the host would open one door and it appears to have a goat behind it. After revealing what’s behind the door, the host asks you if you want to change your door.

After the player chooses a door, the host would open one door and it appears to have a goat behind it. After revealing what’s behind the door, the host asks you if you want to change your door.

The golden age question is: Will you switch or not?

When the door with the goat was revealed, majority of people assume that their chance of winning changed to 50%. Hence, they think that switching their chosen door does not make any difference. However, switching your chosen door doubles your probability of winning. Here is why:

From the beginning, the probability of choosing the car is 1/3, while the probability of choosing the goats is 2/3. This means that it is more likely to choose a door that has a goat behind it. After the host reveals the door with a goat behind, it does not mean that your chances of winning becomes a 50-50.

In concept, we make use of probabilities because there are experiments that do not have a certain outcome, but the chances of each outcome could be calculated. Since the outcome is not certain, the chances of each are random. For example, flipping a coin is random because there is no certainty that we will either get a head or a tail. Additionally, the events of getting a head or a tail is independent from each other.

Going back to the game show, the event of you choosing the door with the car is random because the object behind each door is not known. The chance of getting a car is 1/3 because it is behind one of the doors. However, the chances did not change to 1/2 just because the host opens one door. It is given that the host will open a door with a goat behind it. If he were to reveal the grand prize on the first reveal, then the suspense of the game show will be ruined. If that is true, why is the chance not 1/2 if there are only two doors with one that has a car behind it? This is because of the concept of drawing without replacement.

Drawing without replacement happens when there is a set of choices, and one must choose more than once but it was not done simultaneously. This means that after choosing, the person must choose again. This changes the chances because the sample is now smaller and depending on what the person picked, the probability of certain options may change.

For example, there is a basket full of different colored balls. The colors are red, blue, and green. There are four of each color. If the player has 2 chances of taking a green colored ball and he failed at the first try, his chances will increase on the second try. From a 33% (1/3) chance to 36% (4/11).

Using the concept of drawing without replacement, the chances should change from 1/3 to 1/2 but this is not as simple. This is because the chances of taking a goat from before was 2/3. When one door is opened with the goat, the door the player chose now carried the 1/3 chance of having a goat from the opened door. This means that, the chosen door now has a 2/3 chance of having a goat behind it.

Using the Complement Rule of Probability where 1 is subtracted by a chance of an event to get the probability of the other event, the chances of the other door could be calculated. This would mean that the other door that the player has not selected has a 1/3 chance of having a goat behind it because the door that the player chose has the 2/3 chance. Since the door that was not selected has a 1/3 chance of having a goat, this means that it has 2/3 chance of having a car. From a 1/3 chance of getting a car, if the player would switch doors, it will become 2/3 because the host opened a door with a goat in it making the chances double.

With that, we can say that switching doors gives us a better chance of winning because you would know that the host is avoiding the grand prize, making your chances higher. This does not guarantee winning a car because there is still a 1/3 chance that the switched door has a goat behind it. However, this doubles the assurance that the door has a car behind it.

If you still don’t get it, we can use a table to show how there is a 2/3 percent chance of winning the prize when you switch doors.

\[ \begin{array}{|c|c|} \text{Chosen Door} & \text{Prize Door} & \text{Revealed Door} & \text{Switch} & \text{Didn't Switch}\\ 1 & 1 & 2/3 & \text{Loss} & \text{Win}\\ 1 & 2 & 3 & \text{Win} & \text{Loss}\\ 1 & 3 & 2 & \text{Win} & \text{Loss}\\ 2 & 1 & 3 & \text{Win} & \text{Loss}\\ 2 & 2 & 1/3 & \text{Loss} & \text{Win}\\ 2 & 3 & 1 & \text{Win} & \text{Loss}\\ 3 & 1 & 2 & \text{Win} & \text{Loss}\\ 3 & 2 & 1 & \text{Win} & \text{Loss}\\ 3 & 3 & 1/2 & \text{Loss} & \text{Win}\\ \end{array} \]

Total wins when you switch: 6/9

Total wins when you don’t switch: 3/9

The table above lists all the possible combinations on how the event can occur. The results tell whether you win more likely in switching or not. Based from the results, you are more likely to win the grand prize by switching.

To summarize, you are right 1/3 times when choosing your door. That means that you are more likely to choose wrongly because chances are, you are going to choose the door with the goat 2/3 times. Revealing one door only gave you more information about the remaining doors. It just means that one door has the prize and the other has the goat, but because you picked wrong most of the time in your first choice, you are more likely to win the prize when you switch.

Ultra Lotto 6/58

Lottery is a form of a gambling system, a game of chance, as well as a method of raising money for a charitable purpose, where people buy numbered tickets, and prizes are given to those who are holding the winning numbers dictated by a random drawing system.

One of the largest state lottery companies is PCSO or Public Charity Sweepstakes Office. They host one of the largest nationwide lottery games, and the money that they generate is obligated to serve as funds for the social welfare of the country. They currently hold different games varying in gameplay and combinations.

Their most recent and biggest one yet, is the Ultra Lotto 6/58. Like some of their other games, the concept is pretty simple. The draw mechanics would be the same, but six numbers are chosen between 1 to 58 without replacement. The lottery ticket that contains the six drawn numbers in any order would claim the grand prize.

Speaking about lotteries, everyone dreams of reaching there goals and passion one day, and then there are people who just wants to win the lottery to be an instant millionaire. Then that day came, all hopes are lost and you’re just sitting there in your room, thinking that, what if you just bet on all of the possible combinations in the Ultra Lotto 6/58, then you’re already settled for life right? That is not the case.

Using the combination formula:

\[C(n,r)=\dfrac{n!}{r!(n-r!)}\]

we let \(r=6\), which is the number of winning balls out of \(n=58\), which is the number of total balls.

Substituting the given values would yield:

\[C(58,6)=\dfrac{58!}{6!(58-6)!}=40,475,358\]

From solving, there would be 1 out of 40,475,358 possible combinations with 6 digits you can make out of the numbers between 1 and 58.

That is a lot of possible combinations, and with each lottery ticket costing about Php 20 each, you will be spending roughly about \(20 \times 40,475,358= 809,507,160\) pesos if you plan to bet on every single one.

So, even if you win the grand prize of Php 50 million, you still lose a large sum of money - Php 759,507,160 to be exact - which is never a good thing as it will not be enough to cover all the expenses you wasted from the start.