# Probability distribution for a random variable related to self-assignment of a person's degree of belief

I am not a psychologist or neuroscientist. I am a mathematician wondering about how a person normally evaluate its own degree of belief in a given proposition.

Suppose a person X is given a question in the following format:

Assign your degree of belief in the proposition P. (a) Certainly true. (b) Probably true. (c) I don't know. (d) Probably false. (e) Certainly false.

This person knows that he/she will gain a score S, for this question, defined by:

• If X answered (a): S = +10 if P is true; S = -10 if P is false.
• If X answered (b): S = +5 if P is true; S = -5 if P is false.
• If X answered (c): S = 0 if P is true; S = 0 if P is false.
• If X answered (d): S = -5 if P is true; S = +5 if P is false.
• If X answered (e): S = -10 if P is true; S = +10 if P is false.

Given that the expected value for the score S that X will gain is a known number E, I ask:

• What is the probability of X to answer (a)?
• What is the probability of X to answer (b)?
• What is the probability of X to answer (c)?
• What is the probability of X to answer (d)?
• What is the probability of X to answer (e)?

In short: Supposing that we know the expected value of the score S that X will gain, what is the best model (taking into account facts of human psychology) for the probability distribution of the random variable "degree of belief that the person X assigns to the proposition P"?