**Discrete distribution**

When the values of a variable can be listed, the variable is discrete. These can include, for example, grades, native languages, place of residence, or any variable that can be listed.

A continuous variable is, for example, age or time or even the area of the apartment. Such variables take on an infinite number of values, and not all possible values can be listed. The closer the issue can be measured, the more accurate the values of the variable will be. Can you list all possible real numbers between the numbers 0 and 1? If you choose any two adjacent numbers, there are always numbers between them.

**Example 1**

There were 30 pupils in the class, of which 8 pupils were native speakers of Finnish, 7 pupils were absurdists, 10 pupils were Gothic and five were Akkadians.

The spoken mother tongues are presented with a suitable diagram. Native languages can be easily listed, so a bar chart fits this well.

**Example 2**

By throwing a regular tetrahedron shaped dice, you can roll a number 1, 2, 3, or 4. The probability of these are equal. A player simultaneously rolls a tetrahedron and a regular dice and calculates the sum of the numbers.

a) Find the probabilities of the sums of all possible outcomes.

b) Find the expected value of the sum of the numbers.

(YO K2014 Long Mathematics)

Solution

We put all sums in a table

We have 24 different outcomes

a) The table gives the probabilities

b) The expected value of the sum is (The expected value is discussed below.)

It can also be calculated as follows. The expected value of the number of a regular dice is 3,5 and the expected value of the number of a tetrahedral dice is 2,5, so the expected value in the question is obtained as the sum of these.

**Example 3**

The sum of the numbers of the dice in the previous example is discrete. The distribution of probabilities of sums is presented as a bar graph.

### Expectation value and standard deviation

The expected value is the expected value of the variable. The expected value is calculated as the sum of each value multiplied by the corresponding probability and these are added together.

The expected value is denoted with a capital letter E.

**Example 3**

The probabilities of the three strikers of the Ringette team to score a goal with a penalty shot are 65%, 75% and 54%. Each of the three attackers gets one attempt.

a) What is the probability that at least one attacker will score a goal?

b) Calculate the expected value of the number of penalty shots.

(YO K2012 Long Mathematics)

Solution

A complementary event is used. The complementary event to the "At least one goal" event is "no goals".

**P (at least 1 goal) = 1 - P (no goals)**

**P (no goals) = (1 - 0,65) (1 - 0,75) (1 - 0,54) = 0,04025,**

**P (at least 1 goal) = 1 - 0,04025 = 0,95975 ≈ 0,96.**

b) Let P(n) be the probability that n goals will be scored.

Then the expected value is

**P(1) = 0,65(1 − 0,75)(1 − 0,54) + 0,75(1 − 0,65)(1 − 0,54) + 0,54(1 − 0,65)(1 − 0,75) = 0, 24275,**

**P(2) = 0,65 · 0,75(1 − 0,54) + 0,65 · 0,54(1 − 0,75) + 0,75 · 0,54(1 − 0,65) = 0,45375**

**P(3) = 0,65 · 0,75 · 0,54 = 0,26325.**

Answer: a) 0,96, b) 1,94. (That is, in this case, the expected number of goals would be 2)

### Standard deviation

This measures the magnitude of the variation in the values of the variable compared to the expected value.

Calculate the standard deviation of the number of goals in the previous example.

**Turn on the subtitles if needed**

**Exercises**

1. We roll two dice and add up the sums of the numbers. Find the probabilities and expected value of the sums.

2. Liisa-Petter answered an exam with eight multiple-choice questions. She got one point for a correct answer and -0,5 points for a wrong answer. The questions had four possible answers, one of which was correct. Liisa-Petter answers the test by guessing. What is the expected value of her score?

3. Marja-Unto played a game where four numbers out of ten were chosen. One round cost 2 euros. If she correctly chose all four, she won €10. Three correctly chosen won 5 euros. Two correctly chosen won 2 euros and one got a euro. Calculate the expected value for the winning amount of one round. Evaluate Marja-Unto's winning situation after a hundred rounds.

4. In one computer game, the player advances to the top level according to the diagram below and gets a score marked on the diagram. At each intersection, he randomly selects one of the equal options and advances to the next level.

a) What is the probability that a player will achieve the highest score of 40?

b) Find the expected value of the score.

(YO2011S Long Mathematics)