"Officieel antwoord" met behulp van kansrekening:
The answer is 0. Here's how to look at this.
We have a sample space (the set of all possible choices) which is S={A,B,C,D}.
We have a probability measure P defined on S, namely, P(A)=P(B)=P(C)=P(D)=0.25.
We have a random variable X which is a function from S to R (the set of real numbers).
X is defined as X(A)=.25, X(B)=.5, X(C)=.6, and X(D)=.25.
An event is a subset of S. For example the event that .2 < X < .55 is the event of
making a choice where the answer is a number between .2 and .55. This occurs if you
choose A or B. So the event is the subset of S given by {A,B}. The probability of an
event is just the sum of the probabilities of the elements in the event. So The
Probability of making a choice where the result is a number between .2 and .55 is
P(A) + P(B) = .25 + .25 = .5. In more compact mathematical notation this would be
given as: event Q={ s in S : .2 < X(s) < .55}={A,B}. And P(Q)=.5. (The Q is just an
arbitrary letter assigned to this event)
For any real number c we can consider the event Q(c) = {s in S : X(s)=c}, that is
that we made a choice where the answer was c. For example if c=0.17, Q(c) is the
empty set since no s in S gives X(s)=.17. The probability of the empty set is always 0.
So P(Q(c))=0 in this case.
In Raymond's question "being correct" is the event Q(c) for which P(Q(c))=c, and the
"chance that you will be correct" is P(Q(c)) which equals c. So we need to find the number
c for which P(Q(c))=c.
Let's evaluate P(Q(c)) for all the possible values of c.
Q(.25) = {A,D} so P(Q(.25)) = .25 + .25 = .5
Q(.5) = {B} so P(Q(.5)) = .25
Q(.6) = {C} so P(Q(.6)) = .25
Q(c) = the empty set for any other value of c, so P(Q(c))=0 for any other value of c
and in particular P(Q(0)) = 0.
So we have answered the question : for what value of c is P(Q(c))=c, namely c=0
