Exam Questions - Block 1: Subjective Probability

[SebastianVeuskens]

Subjective probability

[SebastianVeuskens]

Question 1

Main characteristics

• It is not a physical property of an event
• It rather formalizes a believe of a person about an event
• It is subjective, since it describes the relation between the individual and the world

Mathematical representation
The subjective probability of an event E is the price P(E) the individual considers fair for the following bet:

1. In case of event E, receive 1 back
2. In case of not event E, receive 0 back

A probability P(E) is coherent if and only if it holds:

0 <= P(E) <= 1

Not coherent means that, depending on the price, a certain strategy in betting always leads to a certain win. This is the case only for negative probabilities and probabilities greater than 1.

[SebastianVeuskens]

For this question I want to propose some definition taken by the Advanced Probability which you can find the direct source at this link) course since I think they are more comprehensible.

The probability of an event, $P(E)$ is the price that a coherent person allows to pay in a bet for winning 1€ provided that the event $E$ is true.
This means that, as @SebastianVeuskens propose, that the bet follows this scheme:

• Provided $E$ is true, you win 1
• Provided $E$ is false, you win 0 (you don't receive anything)

What do we mean with the term coherent?
Suppose that $P(E) = 2$ (remember that 2 is the price you pay to play the game/bet): in this case two things can happen

1. If $E$ is true you won 1$ but you paid 2$, so you lost 1$
2. If $E$ is false you won 0$ and you paid 2$, so you lost 2$.
   Clearly, no coherent person, in this sense, would want to play at this game.

Plus

What is the shortcoming of this definition?
Not everyone gives the same probability of an event, even if they are all "coherent", and there is no reason that one opinion is more accurate than the others.