Probability calculus - Stochastics and statistics

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Probability calculus
Stochastics and statistics

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In a probability experiment, the result quantity Ω specifies all possible results that could come out in the experiment.

Examples:

a) Experiment: A card will be picked out of a deck.

Ω = {diamond 7, diamond 8, diamond 9, diamond 10, jack of diamond, queen of diamond, king of diamond, ace of diamond, hearts 7, hearts 8, hearts 9, hearts 10, jack of hearts, queen of hearts, king of hearts, ace of hearts, spades 7, spades 8, spades 9, spades 10, jack of spades, queen of spades, king of spades, ace of spades, clubs 7, clubs 8, clubs 9, clubs 10, jack of clubs, queen of clubs, king of clubs, ace of clubs }

b) Experiment: A dice is thrown and the outcome is noted.

Ω = {1, 2, 3, 4, 5, 6}

c) Experiment: A dice is thrown and the number of times that an odd or even number is thrown is noted.

Ω = {even, odd}

d) The dice is thrown twice and the sum of both is noted.

Ω = {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}

Attention: The results here are not equally probable!

If all results in an experiment are equally probable, it is called a Laplace-Experiment.

In the probability theorem, the result is a quantity (if necessary formulated orally). If the result is probable, than the quantity is a part of the quantity Ω.

Example:

1. Experiment: the dice is thrown once.

2. Ω = {1, 2, 3, 4, 5, 6}

3. Results

Result A: “A “7” is thrown” A = {7} Probability of A = P (A) = 0

Result B: “An even number is thrown” B = {2, 4, 6} àP (B) = 0,5

1. Step: define: definition of an experiment

2. Step: notation of Ω

3. Step: definition and notation of the result A, B…

4. Step: calculation of the probability of the results. Only in the Laplace experiment counts: P (A) =

The opposite result to a result A is defined with. It contains the entire elements that are not A. This means, = Ω \{A}.

Example: A simple coin throw Ω = {W, Z}. A = {W}; = Ω \{W} = {Z}.

For the probability of a result and the opposite result counts: P (A) + P () = 1 = P (Ω) or/and P () = 1 – P (A).

The tree diagram is the instrument used to calculate the probability, even the most complicated results!

Example: Experiment: 2 blue and 6 green marbles are in an urn. Three of those marbles are picked out without being put back.

There are also possibilities to calculate the probability differently. As an example, the lottery game can be used. In a lottery, 6 marbles are picked without being put back from 49. The goal is to become as many equal numbers (hits) as possible.

To simplify the tree diagram:

The probability that all 6 numbers are the same can be calculated:

∙ the number of paths that are possible when wanting 6 of the same numbers (this is just one path) = 7,151123042∙10^-8

The probability that 5 numbers are the same:

= 1,844989951∙10^-5.

Why are there 6 paths for 5 hits? In other words, how many possibilities can be distributed on 6 spots for a strike (no hit)?

T T T T T N
T T T T N T
T T T N T T
T T N T T T
T N T T T T
N T T T T T

The probability for 4 hits:

= 9,686197244∙10^-4

Why 15 paths? For the first strike there are 6 possabilities to distribute it on 6 spots, for the second strike, there are only 5 left. All combinations would then be 6 ∙ 5 = 30. The combinations where only spots are switched are already included though. These are double, since they do not distinguish themselves from another.

Regularity can be observed. For 3 hits (3 strikes) there will be paths, for 2 hits (4 strikes) paths, for 1 hit (5 strikes) paths and for 0 hits paths. To calculate this, the probability and the number of paths are multiplied.

A product out of factors that goes backwards from n to 1 (so from n …∙ 2 ∙ 1) is called n-factorial. Written: n! The factorial of 0 (0!) is defined as 1. The factorial is only an abbreviation.

The abbreviation for the number of paths:

1 hit for 6 spots:

2 hit for 6 spots:

3 hit for 6 spots:

4 hit for 6 spots:

5 hit for 6 spots:

6 hit for 6 spots:

Generally:

“n over k” is the abbreviation for the before written formula.

Probability 3 hits

= 0,017650403

Probability 2 hits

= 0,132378029

Probability 1 hits

= 0,41301945

Probability 0 hits

= 0,435964975

Another example: Poker

The probability that one becomes 4 aces in the first hand (every player has 5 cards, the game is played with 52 cards, every card is represented 4 times Ω = {4 x 2, 4 x 3, …, 4 x king, 4 x mal ace}).

P (x = 4) = = 1,846892603 ∙ 10^-5

Bernoulli experiment

A probability experiment with exactly two possible results (hit or strike), that is performed n times (independently from another) is called an n ray Bernoulli experiment.

Example: double coin throw

Probability of seeing heads twice:

P (x = 2) = = 0,25