Classical definition of probability. Properties of probability. Calculating probability of events

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Klasyczna definicja prawdopodobieństwa. Własności prawdopodobieństwa. Obliczanie prawdopodobieństw zdarzeń losowych

Learning objectives

You will learn concepts related with calculating probability and you will learn properties of probability.

Learning effect

Learning concepts related with calculating probability and learning properties of probability.

Prepare information for the following subjects:

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nagranie abstraktu

I. Experiments, elementary events, sample spacesample spacesample space, operations of events.
II. Identifying probabilityprobabilityprobability.
III. Properties of probability.
IV. Classical probability.

See if information you prepared are among the following one:

I. Experiments, elementary events, sample space, operations of events

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nagranie abstraktu

An experimentexperimentexperiment – a repeatable experiment whose outcome we cannot predict.
An elementary eventelementary eventelementary event – an outcome of the experiment.
Sample space – a set of all elementary events, marked with the letter Ω (omega).
A complementary event to the event Acomplementary event to the event Acomplementary event to the event A – an event A’ to which are favourable all elementary events that are not favourable to the A event.

II. Identifying probability

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nagranie abstraktu

Probability defined on a finite sample spacesample spacesample space Ω is such function P that assigns to each event A, A ⊂ Ω a real number P(A) in such a way that:
- (A1) P(A) ≥ 0
- (A2) P(Ω) = 1
- (A3) if A, B ⊂ Ω i A ∩ B = ∅, then P(A ∪ B) = P(A) + P(B).
A pair (Ω, P) – probability space.

III. Properties of probability

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nagranie abstraktu

Properties of probabilityprobabilityprobability:

P(∅) = 0,
if A ⊂ B, then P(A) ≤ P(B),
P(A) ≤ 1,
P(A') = 1 - P(A),
P(A ∪ B) = P(A) + P(B) - P(A ∩ B),
If events AIndeks dolny 1, AIndeks dolny 2, …, AIndeks dolny n ⊂ Ω are mutually exclusive then P(AIndeks dolny 1 ∪ AIndeks dolny 2 ∪ ... ∪ AIndeks dolny n) = P(AIndeks dolny 1) + P(AIndeks dolny 2) + ... P(AIndeks dolny n).

IV. Classical probability

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nagranie abstraktu

Classical probabilityprobabilityprobability definition.

If the sample space Ω is finite and all elementary events are equally probable and $A$ is any event in this space, then:

P (A) = \frac{|A|}{|Ω|}

Task 1

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nagranie abstraktu

Open the interactive illustration, that describes how to calculate probability of an event and watch it carefully.

We calculate the probability of an event A where after rolling a symmetric, six‑sided dice and a flip a symmetric coin, we obtain an even number on the dice and heads.

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Probability of an event

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We roll the dice and flip the coin and get a set of elementary events:

Ω = {O,1, O,2, O,3, O,4, O,5, O,6, R,1, R,2, R,3, R,4, R,5, R,6}
|Ω| = 12
A – event A is an event where we can heads an an even number
|A| = 3

Probability of the event A: $P (A) = \frac{3}{12} = \frac{1}{4} = 0, 25$

Probability of the event A is 0,25.

Task 2

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nagranie abstraktu

We flip a symmetric coin three times.

a. Write all elements of the sample spacesample spacesample space Ω.
b. Write all elementary events favourable to events A and B if:

A - the event where we get at least one heads,
B - the event where we get heads three times or tails three times.

Give number of elementary events favourable to events A and B.

Task 3

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nagranie abstraktu

In a certain experimentexperimentexperiment, the set of elementary events is:
Ω = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}.

The set of elementary events favourable to events A, B and C is defined as follows:
A = { x: x ϵ Ω ʌ x ≤ 4},
B = { x: x ϵ Ω ʌ 3 ≤ x ≤ 5},
C = { x: x ϵ Ω ʌ 4 ≤ x ≤ 8}.

Write all elementary events favourable to events:
A, B, C, A ∪ B, A ∩ C, C’.

Task 4

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nagranie abstraktu

A random letter was chosen from letters of the word PRAWDOPODOBIEŃSTWO. Calculate probabilityprobabilityprobability that it is letter P or O.

Task 5

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nagranie abstraktu

Knowing that $\frac{P (A)}{P (A')} = 5$ , calculate $P (A)$ and $P (A')$ .

Task 6

An extra task:

Calculate P(A ∪ B) and P(A’ ∩ B’), knowing that

A ⊂ Ω and B ⊂ Ω are mutually exclusive events,
P(A') = $\frac{2}{3}$ ,
P(B') = $\frac{1}{2}$ .

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nagranie abstraktu

An experimentexperimentexperiment – a repeatable experiment whose outcome we cannot predict.
An elementary eventelementary eventelementary event – an outcome of the experiment.
Sample space – a set of all elementary events, marked with the letter Ω (omega).
A complementary event to the event Acomplementary event to the event Acomplementary event to the event A – an event A’ to which are favourable all elementary events that are not favourable to the A event.
Probability defined on a finite sample spacesample spacesample space Ω is such function P that assigns to each event A, A ⊂ Ω a real number P(A) in such a way that:
- (A1) P(A) ≥ 0,
- (A2) P(Ω) = 1,
- (A3) if A, B ⊂ Ω i A ∩ B = ∅, then P(A ∪ B) = P(A) + P(B).
Classical probabilityprobabilityprobability definition. If the sample space Ω is finite and all elementary events are equally probable and $A$ is any event in this space, then: $P (A) = \frac{|A|}{|Ω|}$ .

Exercises

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Exercise 1

We roll a symmetric, six-sided dice twice. Probability of getting the same number both times is equal to:

$\frac{1}{36}$
$\frac{1}{6}$
$\frac{1}{2}$
36

zadanie

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Exercise 2

We know about the A and B events that A ⊂ Ω and B ⊂ Ω and also P(A ∪ B) = 0,5 and P(A) = P(A ∩ B) = $\frac{1}{3}$ . Calculate P(B) i P(B – A).

P(B) = 0,5

P(B – A) = $\frac{1}{6}$

Exercise 3

We draw one number from a set of two‑digit numbers. Calculate the probability that this number can be divided by 2 or by 3. Write the solution and answer in English.

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Exercise 4

Indicate which pairs of expressions or words are translated correctly.

prawdopodobieństwo - probability
doświadczenia losowe - experiment
zdarzenie elementarne - elementary event
przestrzeń probabilistyczna - elementary event
doświadczenia losowe - probability

zadanie

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Match Polish terms with their English equivalents.

zdarzenie przeciwne do zdarzenia A
doświadczenia losowe
sample space
an experiment
a complementary event to the event A
zdarzenie elementarne
an elementary event
przestrzeń probabilistyczna

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Glossary

complementary event to the event A

zdarzenie przeciwne do zdarzenia A

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wymowa w języku angielskim: complementary event to the event A

elementary event

zdarzenie elementarne

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wymowa w języku angielskim: elementary event

experiment

doświadczenia losowe

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wymowa w języku angielskim: experiment

probability

prawdopodobieństwo

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wymowa w języku angielskim: probability

sample space

przestrzeń probabilistyczna

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wymowa w języku angielskim: sample space

Keywords

elementary eventelementary eventelementary event

experimentexperimentexperiment

sample spacesample spacesample space

Scenariusz

Exercises

Glossary

Keywords