-------- Story --------

(From Bertsekas and Tsitsiklis's Introduction_to_probability book)

A patient is admitted to the hospital and a potentially life-saving drug is administered.
The following dialog takes place between the nurse and a concerned relative.

RELATIVE: Nurse, what is the probability that the drug will work?

NURSE: I hope it works, we’ll know tomorrow.

RELATIVE: Yes, but what is the probability that it will?

NURSE: Each case is different, we have to wait.

RELATIVE: But let’s see, out of a hundred patients that are treated under similar conditions,
how many times would you expect it to work?

NURSE (somewhat annoyed): I told you, every person is different, for some it works, for some it doesn’t.

RELATIVE (insisting): Then tell me, if you had to bet whether it will work or not, which side of the bet would you take?

NURSE (cheering up for a moment): I’d bet it will work.

RELATIVE (somewhat relieved): OK, now, would you be willing to lose two dollars if it doesn’t work,
and gain one dollar if it does?

NURSE (exasperated): What a sick thought! You are wasting my time!


--------  Analysis  -------

When nurse says " .... for some it works, for some it doesn’t ...."
 she is kind of indicating that there is 50-50 chance of working and not working.

If nurse accepted the bet towards the end, in her mind, she means this: Her pay off would be
(assuming that probabability of drug working is p):

-2 * (1-p) + 1 (p)
 = -2 +2p +p
 = -2+3p

 This has to be > 0 for her not to lose money and be happy
 Equivalently: p > 2/3

 If p = 2/3, she does not lose or gain money: indifferent

 if p < 2/3, she is unhappy (or won't to bet if she has any doubt if p < 2/3)

 And, of course, the it does make sense to assign a probability value for "if the drug works on patient"
 once the result is known. It is either 0 or 1. But, nurse would be still indifferent/sad/happy as long as she is not
 revealed of the result.


------   Summary  --------

So, if the probability of an event = p, means, this for a rational person:

He would be indifferent between the two situations:

1) willing to bet: lose c dollars if event does not happen and gain d dollars if the event happens
     ( expected payoff :  d * p + [ -1 * (1-p) * c ] )
2) not betting at all

Instead of being indifferent, he would be happy if  d * p - c + c  * p > 0
or  p > c / (c + d)

Instead of being indifferent, he would be sad otherwise [ or won't bet ] , i.e., p <  c/(c+d)

Obviously he would be indifferent if p = c / (c+d)

And, of course, once the event takes places (or does not),
P(event) does not make sense as we know the result for sure: 0 or 1. But the rational person's state of
mind is still the same as long as he is not revealed of the outcome.

---------- End of story ---------