Tag: probability

Posted in Science & Nature

Birthday Problem

Posted on by Jinavie

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 Science & Nature

Card Shuffling

Posted on by Jinavie

Take a standard deck of playing cards. Shuffle it thoroughly and set it on the table. Consider this: what is the probability that the order those 52 cards are in is the same as the order of a deck shuffled by someone else? The answer can be found using a simple maths equation: 52!

! denotes a factorial, where you multiply the number to every other positive integer smaller than it. For example, 5! = 5 x 4 x 3 x 2 x 1 = 120. Due to its nature, factorials grow rapidly – even faster than exponentials. For example, 10! is 3.6 million and 15! is 1.3 quadrillion. By 52!, the number grows to:

80,658,175,170,943,878,571,660,636,856,403,766,975,289,505,440,883,277,824,000,000,000,000.

This number is so big that if every star in our galaxy had a trillion planets, each with a trillion people living on it, all shuffling a trillion deck of cards at the rate of 1000 shuffles per second, since the beginning of time, only now would someone have a deck that is in the exact order as your deck.

Ergo, you can say with absolute, mathematical certainty, that the deck you have shuffled is in an order never created by any human being in the history of the world.

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Posted in Life & Happiness

True Love

Posted on by Jinavie

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.

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Posted in Science & Nature

Rock-Paper-Scissors

Posted on by Jinavie

Rock-paper-scissors is a game with a long history. The earliest example of the game is a Chinese game called huoquan, which follows a cyclic rule where the frog eats the slug, the slug dissolves the snake and the snake eats the frog. The reason why rock-paper-scissors has been saved throughout history is because of the uncertainty it contains. Any hand you choose, the chance of winning is the same. Ergo, there is no single best choice and there is no move that will always win. But this is still a game played by people. It is not a game played by emotionless machines, meaning that you can use human psychology, the surfacing of emotion and specific signs and movements to help deduce your opponent’s hand. Mentalist Derren Brown can read tiny flickering of muscles in the opponent and microexpressions to pull off his “undefeatable rock-paper-scissors trick”, but this is near impossible for a normal person to try. However, you can use the following strategies to improve your odds.

  1. Use paper on a beginner: Statistically, people prefer using rock. Males especially have a strong tendency to play rock.
  2. Use scissors on an experienced player: People who know the first trick can be defeated by going one step further.
  3. Use a hand that loses to the hand your opponent played: This uses the psychology of the opponent wanting to mix up hands and wanting to beat the hand you last played (which is the same as theirs as you drew).
  4. Say what you will play and play that hand: In a competitive situation like rock-paper-scissors, people tend not to trust others. Thus, if you say you will play a certain hand, they will think is a trap and not play the hand that defeats that hand. For example, if you said you will play scissors, the opponent will play paper or scissors and you will either win or draw.
  5. Do not give the opponent a chance to think: People have a subconscious tendency to play a hand that beats the hand that they played before. Without time to think, the subconscious takes action meaning that you can predict their move. If you do the same as strategy 3 and play a hand that loses against the opponent’s previous hand, you will win.
  6. Suggest a certain hand: This is a form of hypnosis where you suggest something to the opponent’s subconscious. To use this trick, pretend to go over the rules by saying “rock, paper, scissors” then play a certain hand. The opponent will likely play the hand that the subconscious last saw.
  7. If you keep drawing, use paper: This is the same as strategy 1.

Unfortunately, rock-paper-scissors has an equal probability of a win and a draw, meaning draws are rather common. Thus, a computer engineer called Samuel Kass devised a game where two additional hands are added: rock-paper-scissors-lizard-Spock. Lizard is played by making your hand into the shape of an animal’s head, while Spock is played using the Vulcan Salute from the science fiction show Star Trek, where you make a V-shape with two fingers on each side. The rules are as follows.

Scissors cut paper. Paper covers rock. Rock crushes lizard. Lizard poisons Spock. Spock smashes scissors. Scissors decapitate lizard. Lizard eats paper. Paper disproves Spock. Spock vaporizes rock. Rock crushes scissors.

As each hand has two ways of winning, the odds of winning is 10/25, or 2/5 and the odds of drawing is 5/25, or 1/5. As you can see, you have double the chance of winning compared to drawing, making the game much faster to play than the original game.

Posted in Science & Nature

Monty Hall Problem

Posted on by Jinavie

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?”

Is it to your advantage to change your choice?

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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:

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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

Posted on by Jinavie

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.

Posted in Science & Nature

From Cell To Birth: Fertilisation

Posted on by Jinavie

Once the sperm enters the vagina, the real battle begins. The vagina is highly acidic, an environment in which sperm can only survive 2~3 hours. It is crucial for the sperm to enter the uterus through the cervix, but only 1% of the 200~300 million sperm make it through.

Even within the uterus, they must brace harsh conditions as they travel against gravity. After about 5 hours of intense swimming, the sperm reach the top of the uterus. Here they face a choice: go left or go right. Half the sperm make the wrong choice and head down the eggless fallopian tube and ultimately die. The rest navigate their way through the maze of folds in the fallopian tube, often getting lost or sticking to the wall thinking that it is an egg.

About 200 sperm finally make it to the egg, which sits in the ampulla of the fallopian tube. But as always, there is competition even at this final moment. Only one sperm can win the race, and the fastest one will ultimately produce a new life.

When the first sperm touches the egg, a series of chemical reactions occur, essentially “priming” the sperm. This causes it to start the acrosome reaction, where it releases a hoard of enzymes from its head, digesting away the covering shell (zona pellucida) of the egg. It then becomes supercharged, using all of its energy to drive itself inwards until it reaches the oocyte within. As soon as this happens, the tail breaks off, and one final chemical reaction as the calcium level spikes occurs to release more enzymes that prevent the acrosome reaction in other sperm. It also solidifies the zona, forming an impenetrable shield to prevent other sperm coming in (polyspermy can lead to a failed pregnancy).

The calcium spike that causes the above cortical reaction also triggers the egg to divide, so that it reaches the most mature stage. The winning sperm can then combine its nucleus with the oocyte, forming the 46 chromosomes that will set the genetic basis of the new zygote (first stage of a baby).

To reach the egg, the sperm must travel over 20cm – beating its tail over 20,000 times. The probability that a certain sperm will fertilise the egg is 1 in 500,000,000.
Life starts under a near-zero probability condition.


(Full series here: https://jineralknowledge.com/tag/arkrepro/?order=asc)