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Hi Steemers!
It's Kate.

In this article I would try to explain you the basics of Probability theory and give you an example of its use. That's gonna be interesting!

What does this science study? Please, now take a coin and tell me on which facet it would fall after you throw it? It's not like a theory it is more like some kind of divination. However, if the same coin in the same conditions is thrown for hundreds and thousands of times, there will be a clear regularity that could be described as a law and here is what this science is studying.

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Basic concepts of probability theory**

The probability of the event A is the ratio of the number of favorable outcomes A, to the total number of outcomes that happened during the test:

Let’s practice and try this task:

There are 20 balls in the basket. 15 are red, and others are green. What is the probability to take the green ball?

There are only five green balls.

According to the classical definition of probability, we need to find the total number of possible outcomes, and also find the number of outcomes that are favorable to our event. We can equally likely pull out of the basket any of the twenty balls. Therefore the total number of outcomes is 20.

But we need to find the probability that we take out the green ball and there are only five green balls. Consequently, we have five favorable outcomes.

P (A) = n / N = 5/20 = 1/4 = 0.25

• Any test result is called the outcome, which actually represents the occurrence of a certain event. When you throw the coin there may be to 2 outcomes - eagle or tails.

• The event probability - a numerical expression of the possibility of its occurrence.

Here is a small example:

The test consists of throwing dice, with points (1 to 6). What is the probability that: 1) you’d have 2 points? 2) you’d have an odd number of points?

There are 6 equally likely events (1, 2, 3, 4, 5, 6 points). Therefore, the probability of having any number of points, including 2 through a single toss is .

The appearance of an odd number of points (1, 3 and 5) in the event A is favorable and using our formula we have

The probability of an event is with by a capital letter, and the event itself is taken into brackets, serving as a kind of argument.

For example: - the likelihood of having an "Eagle" after throwing a coin;

The probability can be written as a percentage, but in the case of the probability theory, it is NOT ACCEPTED.

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Operations with events

Please remember the important rule:

Addition operation with events is logical OR, and the multiplication of events is the logical AND.

1. The sum of the two events is an event that one event or another event will happen or both events happen simultaneously. It is written as or .

2. The multiplication of two events is two co-occurred events, in other words, multiplication means that one event and the other event would happen. It is written as or .

There is an interesting example:

Two criminals are being sought. Each of them could be independently detected within 1 day with a probability of 0.5. What is the probability that at least one criminal will be found during 1 day?

The event A is "at least one criminal was found" Let’s divide this event into some simple ones. Let’s take B1 as “the first criminal was found”, and B2 - a second criminal was found. Then, A = B1 + B2 according to the sum of events. Therefore P (A) = P (B1 + B2). Since B1 and B2 – are joint events, then by the theorem about the sum probability events:

P (B1 + B2) = P (B1) + P (B2) - P (B1, B2) = 0.5 + 0.5 - 0.25 = 0.75.

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Generally the probability theory - is a very interesting thing that can be absolutely useful for your different life situations, whether it would be your children’s math homework or the lottery and casino where it is really necessary to have a wide knowledge in mathematical and probability formulas.

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Have you ever heard about the Monty Hall problem?**

The theory of probability can not only develop your mind but also help in different life situations, e.g. if you participate in ques or bet about something, you can increase your chances of the win if you use correct strategy.

Now Imagine that you participate in a TV show and you have to choose one door out of three – there is a car behind one door and goats behind two others. You pick the 1st(where the car is standing) door but the host, who knows what's behind each door, opens another door, let it be the 3rd, and there is a goat. Then he offers you to change your initial choose saying that probably car is behind another door. Would you change your choice? (write answer in comments).

Just watch this amazing excerpt of the "21" movie that answers this question.

As you could notice, in this situation you should change your choice in order to increase your chances and win the quiz. This effect is called Monty Hall problem and it's not obvious from the first glance.

Stay tuned and I’ll tell you more about this effect in my next article.