Posted in Science & Nature

## Birthday Problem

How many people do you need in a room until there are two people with the same birthdays? The pigeonhole principle dictates that (excluding February 29) since there are 365 birthdays, 366 people in a room would guarantee two people sharing birthdays. However, this is only the number needed to absolutely guarantee a pairing. Using a neat statistical trick known as the birthday problem (or birthday paradox), we can find that a much smaller number is needed to solve the problem.

Let us assume that every birthday is equally possible (in real life, some birthdays are more common than others). If there are 30 people in the room, Person 1 has a chance of sharing a birthday with each of the other 29 people (possible pairs). Person 2 can be paired with 28 people (since they have already been “paired” to 1), Person 3 with 27 people and so forth. Therefore, the number of chances are: 29 + 28 + 27+ … + 1. Using Gauss’s handy addition trick, the total number is (29 + 1) x 29/2 = 435. We can see already that although the total number of individuals is only 30, the total number of pairs already exceeds 365. Since the probability of having a certain birthday is 1/365, it is likely that it would occur when you have so many possible chances.

Using statistical analysis, it can be found that when there are 23 people, the odds of there being a match surpasses 50%, making it more likely that two people share a birthday than not. By 70 people, the probability of a match grows to 99.9%. Therefore, with only 19% of the number required by the pigeonhole principle, the birthday problem can say with 99.9% certainty that there will be two people sharing a birthday.

Posted in Simple Pleasures of Life

## Simple Pleasures of Life #23

Happy coincidences.

Life is full of chance and coincidences. Coincidences can range from something simple like bumping into someone you haven’t seen in a while at a supermarket, to what some people call “fate” or “miracle” or “destiny”.

Mathematics will tell you that coincidences are simply the product of the law of large numbers – that it is merely a statistical event. Psychology will tell you that we are just victims of the regression fallacy. Religious people will tell you that it is an act of god.

For me, happy coincidences are just little events in life that spice things up a bit. I don’t believe in fate or destiny or some omnipotent deity, but that doesn’t mean that I treat coincidences as unimportant random events. I find that rather than dismissing it, identifying it as something good makes my life a little bit happier. Who cares if the reason that all the traffic lights were green might be a sign? Who cares if the reason that the song that JUST happens to fit my mood comes on a shuffled playlist was statistical chance? Who cares if you call meeting your soul partner destiny?

What matters to me in the end is that it happened and it made me happy. I also find identifying the coincidences and the steps that led to it happening quite interesting (because I’m a huge nerd, which I say with pride). For example, if I had decided to stick with my plan of going to Asan Hospital instead of Severance Hospital for my selective this year, I wouldn’t have met the girl that invited me to WKMSO and I wouldn’t have met the awesome people I did, nor would I have had my unforgettable NYC/Vegas adventure.

Maybe the reason it makes me happy is that knowing that if even a tiny detail was changed, say if I was put on a different shift to that person, things would have turned out quite differently. For better or for worse. But the fact is that things happened to happen in the way it did and it led to me having an amazing time.

And that my friends, is a long-winding rant-y explanation to why I love the butterfly effect.

Posted in Life & Happiness

## True Love

Let us assume that everyone has a person they are destined to end up with. Can we calculate the probability of such a fateful meeting between a man and a woman?

Suppose that the woman is X and the man is Y. Firstly, X and Y need to be born as human beings. They cannot be born as a worm or an onion or something. Here, we will say that the total number of species is M and the population number of each species as P (technically this part is forcing it slightly, so we can skip it).

Although the two have to beat ridiculous odds just to start, just being born as human beings is not enough. One must be born with XX chromosomes to be a woman, and the other must be born with XY chromosomes to be a man.

Let us assume that the two were lucky enough to be born as a man and a woman. Next, they must live in the same space. If one lives in some Korean city and the other lives in some American rural village, it is unlikely the two will ever meet.

Even if they did live in the same place, X and Y must have subjective qualities that the other person finds attractive. If they are not interested in each other, nothing will happen even if they did meet. By this stage, we have clearly gone past the scopes of mathematics.

Then let us assume that a man and a woman, who fit each other perfectly and born as people, are living in the same space. We are still missing one variable: time. Even if we took only the 5000 years that civilisations have existed, the odds of the two being born in the same era as similar ages is less than 0.001%.

Species, sex, space, time… Statistically speaking, the chances of a man and a woman beating all of these odds to establish a perfect couple seem nearly impossible. But we can clearly see that “true love” exists all around us. Numbers are just numbers. If you find a person that makes your heart skip a beat when your eyes meet, that makes you feel that the more you get to know them, the more you think you cannot live without them; in essence a person that makes you think “this person is The One”, do not let the person slip away. The scenario of you and that person existing on the same space-time and loving each other is something that verges on the impossible.

There is no treasure as rare as true love. If you have found true love, or believe that you have found it, fight to seize it and do everything in your power to protect it. That is the greatest accomplishment you can make in life.

Posted in Science & Nature

## Monty Hall Problem

Imagine that you are on a game show and you are given the choice of three doors, where you will win what is behind the chosen door. Behind one door is a car; behind the others are goats, which you do not want. The car and the goats were placed randomly behind the doors before the show.

The rules of the game show are as follows:

• After you have chosen a door, the door remains closed for the time being.
• The game show host, Monty Hall, who knows what is behind the doors, opens one of the two remaining doors and the door he opens must have a goat behind it.
• If both remaining doors have goats behind them, he chooses one at random.
• After Monty Hall opens a door with a goat, he will ask you to decide whether you want to stay with your first choice or to switch to the last remaining door.

Imagine that you chose Door 1 and the host opens Door 3, which has a goat. He then asks you: “Do you want to switch to Door 2?”

Most people believe that as an incorrect option (goat) is ruled out, their odds of winning the car go up from 1/3 to ½ even by staying on the same Door 1 and there is no benefit to switching. However, it is better to switch doors as this will double your odds of winning the car. To illustrate this point, the following three scenarios (with the car being behind Door 1, 2 or 3) can be imagined, using the above rules of the game:

In Scenario 1, you have already chosen the car (Door 1) so Monty Hall will randomly open Door 2 or 3. Switching will obviously lead you to losing the car. The chance of you losing after switching, therefore, is 1/6 + 1/6 = 1/3 (as either Door 2 or 3 could be opened)

In Scenario 2 and 3, because you chose the wrong door (goat) and Monty Hall will open the door with the goat behind it, switching will lead you to choosing the car (no other choices). As the odd of either scenario happening is 1/3 each, your odds of winning after a switch is 2/3 – double the odds of winning after not switching (1/3, the odd of your first guess being right).

Of course, this is only under the assumption that the rules of the game were followed and that Monty Hall will always open a door with a goat behind it. This problem and the answer suggested was extremely controversial as tens of thousands of readers refused to believe that switching could be a better choice. However, as the above illustration shows, the Monty Hall problem is a veridical paradox – a problem with a solution that appears ludicrous but is actually proven true by induction.

Posted in Science & Nature

## Duel

Three gunslingers called Good, Bad and Ugly duel to the death. They each stand an equal distance from each other and shoot at the same time. Good’s accuracy is 30%, Ugly’s accuracy is 70% and Bad’s accuracy is 100%. Who has the highest chance of survival?

Common sense dictates that Bad, with the highest accuracy, will have the highest survival rate. However, when the duel begins, the following scenario will occur.

Good’s most rational decision is to shoot Bad rather than Ugly. Reason being, shooting the person with the higher accuracy improves your survival rate in the next round. Ugly also chooses to shoot Bad instead of Good as it is the best choice. Lastly, Bad shoots Ugly instead of Good. This scenario can be explained by the following diagram:

Thus, the probability of Bad being alive after the first round is (1-0.3)(1-0.7)=0.21, or 21%. This is because Ugly is killed by Bad on the first shot. On the second round, the probability of Good dying is the same as Bad’s survival rate of the first round, which is 21%. Therefore, Good’s survival rate is 79%. On the other hand, Bad’s survival rate becomes 0.21(1-0.3)=0.147, or 14.7%

Ultimately, the survival rate of each shooter is: Ugly 0%, Bad 14.7%, Good 21%, making Good the most likely winner. This illustrates the fundamental principles of game theory – an extremely useful theory that helps predict the many choices we make in life.