Mathematics: Applications of Probability Theory

By | Thursday, July 11, 2019 4:03

Have you ever heard the word “probability”? A simple example of probability is when we play with coins that have two sides. Try throwing the coin up and catch it again. Then, which side is facing us? How much probability do we have for each side to facing us?

From the example above we can conclude that probability are possibilities that can occur or “degree of certainty” of an event. The theory of probability systematically emerged in the 17th century. At that time Blaise Pascal, Pierre de Fermat, and Antoine Gombaud (also called Chevalier de Méré) were discussing a problem about throwing dice.

Chevalier de Méré asks, “Which is more feasible, getting 6 in 4 times throwing 1 dice, or getting twice the number 6 in 24 throws 2 dice?

Many people think that getting twice the number 6 out of 24 throws is more likely because we can do more throws. From this question they made the calculations described below.

1 dice throw:

The possibility of not getting 6 in 1 time throwing the dice is 5/6. The possibility of not getting the number 6 in 4 throws is: 5/6 × 5/6 × 5/6 × 5/6 or (5/6) 4 so the possibility of getting a number 6 in 4 throws is 1 – (5/6) 4 = 0.517746.

2 dice throws:

The possibility of not getting 2 points 6 in 1 time throwing the dice is 35/3, while in 24 times the throw is likely to be (35/36) 24. Thus, the probability of 1 time double 6 is 1 – (35/36) 24 = 0.491404.

From our calculations, it turns out that getting 2 times the number 6 of 24 throws has a chance smaller than 4 times throwing the dice to get 1 time number 6.

The coin and dice cases that occur above are types of opportunity cases in the classical approach.

In addition, other approaches to opportunities are subjective.

Examples are as follows. Tony is a director at a private company and he is opening job openings for managerial positions.

There are 4 prospective managers who are equally smart and can also be trusted. The highest probability that can be appointed as a manager is determined subjectively by Tony.

From this case the opportunity is an index or value that has limits ranging from 0 to 1.

  • If the opportunity (P) = 0, the probability of occurring is impossible.
  • If P = 1, the chance of an event is certain to occur.
  • If 0 <P <1, the chance is possible.

Another example of an probability is when playing play a deck of card. Have you ever played cards? In a set of playing cards there are 52 cards.

If we want to take a king card in one take and ace in the second take, what are the probability we have? The condition of the card on the first withdrawal not returned back to the deck.

Initial probability to get a king card (X):

P (X) = 4/52 (in a set of playing cards there are 4 king cards):

Probability of taking ace (Y):

P (Y) = 4/51 (in a set of playing cards there are 4 aces):

probability for the above events are:
P (X × Y) = P (X) × P (Y) = 4/52 × 4/51 = 0.006.

Another exciting case about opportunities is the story of a monkey with its typewriter.

Alfred is a monkey who lives in a zoo. He has a typewriter equipped with 26 character letters, 1 space, 1 comma, 1 point, and 1 question mark. Alfred doesn’t understand the language so he can only type randomly. Each character takes 1 second.

The question is, how long does Alfred need to be able to write the word “alfred” from the random typing process.

To calculate this, we can imagine that the word “alfred” can be written into 6 slots: | a | l | f | r | e | d |.

Each slot can be filled with 30 characters (26 letters and 4 punctuation marks on a typewriter).

So, the number of ways to write different characters in 6 slots is: 30 × 30 × 30 × 30 × 30 × 30 = 306 = 729,000,000.

If Alfred types one character every second, it takes more than 23 years to be able to write “alfred” by chance. Wow, long enough, huh.

If the number of slots is raised to 8, Alfred will take 20,000 years!

Knowledge about the basic principles of this probability makes it easier for us to make important decisions later. The concept of this probability has been widely used and become the basis of statistical science. Modern life will not work without it. Risk analysis, sports, sociology, psychology, design and engineering, up to finance highly depend on probability theory.

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