Down with uncertainty, or How to find the probability

Like it or not, but our life is full of all sorts of accidents, both pleasant and not so. Therefore, each of us would not be bothered to know how to find the probability of an event. This will help make the right decisions under any circumstances that involve uncertainty. For example, such knowledge will be very useful when choosing investment options, assessing the possibility of winning a stock or lottery, determining the reality of achieving personal goals, etc., etc.

The formula of probability theory

In principle, studying this topic does not take too much time. In order to get an answer to the question: "How to find the probability of some phenomenon?", You need to understand the key concepts and memorize the basic principles on which the calculation is based. So, according to statistics, the events under investigation are denoted by A1, A2, ..., An. Each of them has both favorable outcomes (m) and the total number of elementary outcomes. For example, we are interested in how to find the probability that an even number of points will be on the top of the cube. Then A is a dice roll, m is a fall of 2, 4 or 6 points (three favorable variants), and n are all six possible variants. The calculation formula itself is as follows:

P (A) = m / n.

It is easy to calculate that in our example the required probability is 1/3. The closer the result to unity, the more likely that such an event will actually happen, and vice versa. Here is a theory of probability.

Examples

With one outcome everything is extremely easy. But how to find the probability, if the events go one after another? Consider this example: from the card deck (36 pcs.) One card is displayed, then it is hidden again into the deck, and after mixing the next one is pulled out. How to find the probability that at least in one case the lady was rushed out? There is the following rule: if you are considering a complex event that can be divided into several incompatible simple events, you can first calculate the result for each of them, and then add them together. In our case, it will look like this: _1/36 + _1/36 = _1/18 . But what about when several independent events occur simultaneously? Then the results are multiplied! For example, the probability that if two coins are dropped at the same time, two tails fall out simultaneously, will be: ½ * ½ = 0.25.

Now let's take an even more complicated example. Suppose we hit a book lottery in which out of thirty tickets ten are winning. It is required to determine:

1. The likelihood that both will be winning.
2. At least one of them will bring a prize.
3. Both will be losing.

So, consider the first case. It can be divided into two events: the first ticket will be happy, and the second will also be happy. We will take into account that the events are dependent, since after each pull-out the total number of variants decreases. We get:

10/30 * 9/29 = 0.1034.

In the second case it will be necessary to determine the probability of a losing ticket and take into account that it can be either the first one on the account or the second one: 10/30 * 20/29 + 20/29 * 10/30 = 0.4598.

Finally, the third case, when the lottery played, even one book can not be obtained: 20/30 * 19/29 = 0.4368.