**Probability**

The history of probability calculus and its beginnings can be seen from correspondence relating to the games of chance between the French mathematicians Pierre Fermat (1601 - 1655) and Blaise Pascal (1623 - 1662). For this historical reason, probability calculus tasks are often related to gambling. So it is worth researching, for example, what kind of cards are in the deck and what a six-sided dice means if they are not already familiar. I do not recommend gambling, at least, not until you can prove it to be profitable.

Event A probability is the number of options favourable to Event A divided by the number of all possible options. It is denoted by P(A), where P comes from the word possibility, and in parentheses is written the event whose probability is calculated.

Probability P can get values between zero (impossible event) and one (certain event).

**Example 1**.

There are four red, six green and three yellow balls in a box.

a) What is the probability that you randomly pick up a red ball from the box?

b) What is the probability that you randomly pick up a brown ball from the box?

c) What is the probability that you randomly pick up a ball out of the box?

**Solution**

a) There are a total of 4 + 6 + 3 = 13 balls

There are 4 favourable options for the event “getting a red ball”

**Answer**: The probability of getting a red ball is about 0,31

b) There are zero alternatives for the event of “getting a brown ball”.

c) There are 13 favourable options for “getting a ball” event.

**Answer: **The probability of getting a ball is 1, which means it is a certain event

**Example 2.** Two six-sided dice are rolled. Calculate with what probability

a) the sum of the faces is exactly seven?

b) the sum of the faces is more than seven?

(c) at least one of the dice shows at least a 5.

We use a table to list all the cases, ie all possible alternatives.

a) In total, there are 6 · 6 = 36 cases, i.e. different alternatives. There are 6 cases favourable for the event “the sum is 7”, as shown

Answer: The probability of getting the sum of the faces exactly seven is 0,17.

b) The number of cases favourable for the event “sum> 7” is 15, so

Answer: The probability that the sum of the faces is more than seven is 0,42

c) The number of cases favourable for the event “one dice is at least 5” is 20, so

Answer: The probability that at least one of the dice is at least 5 is about 0,56

**Example 3**. Marjut, Gunilla and Benedictus, three children from the Upper Ala-Härmälä kindergarten try to line up from the shortest to the tallest. However, they have no idea what this means, but they do know how to stand in a line. What is the probability of their success?

We mark the children with their initials and list all possible options:

MGB, GMB, GBM, BGM, BMG and MBG. So there are a total of six options. Only one of the orders is correct, so

**Answer**: Children stand in the right order with a probability of 0,17

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