What are some examples of subjective probability

Probabilities

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We are now interested in the questionwhat probability has a random event A.

Concepts of probability

There are different ways of defining a probability:

  • classical concept of probability,

  • axiomatic concept of probability (= Kolmogoroff concept of probability),

  • statistical concept of probability,

  • subjective concept of probability.

Classic concept of probability

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First of all forclassical concept of probability (= Laplacian concept of probability), i.e. one counts the elements of the set A and divides this by the number of elements of Ω:

$ P (A) = {\ textrm {#A} \ over {\ textrm {# Ω}}} = {\ textrm {Number of favorable cases} \ over {\ textrm {Number of possible cases}}}. $

Danger: In the following we always write P (A) for the probability of the event A. Some authors write W (A) (with W as probability).

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Example 1:

If you roll the dice twice, you are asked about the probability of rolling a total of 7.

The set A of the favorable elements is A = {(1.6), (2.5), (3.4), (4.3), (5.2), (6.1)} and consists of # A = 6 elements (here “#” is the number sign). The set of all elementary events Ω consists of # Ω = 36 elements, ie the probability for A is $ P (A) = {\ textrm {#A} \ over {\ textrm {# Ω}}} = {6 \ over 36} = {1 \ over 6} $.

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Example 2:

If you toss a coin three times, you are interested in the probability that exactly two heads will fall.

It is Ω = {K, K, K), (K, K, Z), (K, Z, K), (Z, K, K), (K, Z, Z), (Z, K, Z ), (Z, Z, K), (Z, Z, Z)} the set of elementary events, which according to the rules of combinatorics (variation with repetition) from nk = 23 = 8 events exist. The event A, that exactly two heads fall, is denoted by A = {(K, K, Z), (K, Z, K), (Z, K, K)} with # A = 3, the probability for A. so P (A) = P (exactly two heads fall) = $ {\ textrm {#A} \ over {\ textrm {# Ω}}} = {3 \ over 8} $ = 0.375.

Requirements of the classical concept of probability

  • the result set Ω only hasfinite many elements

  • are the elements of Ω, i.e. the so-called elementary eventsequally likely.

These conditions cause difficulties in many experiments. There are situations in which the classical concept of probability cannot be used, e.g. if A does not have a finite number of elements (and thus certainly not Ω) or the elementary events are not equally likely.

Axiomatic concept of probability

One then makes do with that

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Kolmogoroff's concept of probability (= axiomatic concept of probability)

The axioms are

1. P (A) ≥ 0 for all events A

2. P (Ω) = 1

3. P (A1 $ \ cup $ A2 $ \ cup $ ...) = P (A1) + P (A2) + ... for finitely many or for countably infinitely many pairwise disjoint events, i.e. for the A.i $ \ cap $ Aj = Ø if i ≠ j.

NOTE:

  • The third axiom can be found for a finite number of A.i also read as P (A1 $ \ cup $ A2 $ \ cup $ ... $ \ cup $ An) = P (A1) + P (A2) + ... + P (An) for pairwise disjoint events.

  • Aaxiom is a law that isnot prove it leaves. So you have to accept it, everything else can be built on these axioms, i.e. proven. If one does not accept an axiom, i.e. regards it as false, then one must reject everything that can only be proven with this axiom.

Statistical concept of probability

Thestatistical concept of probability can be motivated with an example:

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With a coin it is unknown whether it is fair or not, i.e. whether the probability of heads and tails is exactly equal to ½.

How can you find out what probability the events heads or tails have?

To get an idea of ​​this, you can simply toss the coin 100 times and write down the number of heads and numbers that have fallen. If, for example, exactly 30 heads have fallen, one would use the number 30/100 = 0.3 for P (head) as a guide (not as an exact value!) corresponding to $ {70 \ over {100}} $ = 0.7 for P (number). More precisely: the probability for the occurrence of A is defined as the limit value of the relative frequency of occurrence of this event with an increasing number of repetitions of the random experiment, i.e. P (ω) = $ {\ lim_ {n \ to \ infty} {h_n} (ω )} $.

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Since the coin is merelyfinally often can be rolled, the limit value cannot be calculated in practice. In particular, it is not certain that this limit value actually exists.

On the other hand, the following statement is possible and can be derived from the statistical concept of probability: the probability is the best estimate that can be given for the relative frequency in a long series of random tests.

Subjective concept of probability

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Under thesubjective probability one uses the degree of conviction of a person when a random event occurs, e.g. an expert opinion.

Below is an overview of the various probability terms:

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