Question 1. Should you believe your friend?
We will try to show that the answer is not as obvious as it may seem.
Two investors, Investor I and Investor II are friends and insider traders. Each one of them has access to information not available to the general public. Being friends, they share some information and advice but they are not totally open with each other. Investor I knows an "insider" who can provide him with the value of a quantity X, which to the general public appears to be a random variable. Likewise, Investor II can learn the value of a random variable Y by calling a secret source. Each investor knows what type of information the other one has (for example, Investor I knows that Investor II knows tha value of the random variable Y) but they do not know the information itself (for example, Investor II never tells Investor I the value of Y before it becomes public).
A new investment opportunity appeared on the horizon. If the Congress passes a certain law by the end of the year (let us call this event A) then an investment in a certain company would greatly increase in value. Otherwise, all the money would be lost. The probability of A is 0.5. The potential investment would be break-even if the probability A were 0.6 or greater. One could make a potentially very profitable investment if one could know that the probability of A is greater than 0.6. The value of this probability can be affected by the insider information, of course.
Investor I checked the value of X - it happens to be x1. He did it mostly out of habit, because he knows that X is independent of A. Hence, P(A|X=x1) = 0.5. The insider information is of no help to Investor I.
Soon after that, his friend, Investor II, called him and told him that, in view of his insider information, the investment is definitely profitable, i.e., the conditional probability of A given the true value of Y is greater than 0.6. However, Investor II did not tell Investor I the value of Y.
Question 2. Should Investor I make the investment upon hearing his friend's opinion? (cf. Question 1)
Since Investor II did not tell Investor I the value of Y, the first investor decided to check if the optimism of the second one is consistent with his own information. It turned out that given {X = x1}, the random variable Y can take only two values, y1 or y2. The conditional probabilities happen to be P(A|Y=y1) = 0.7 and P(A|Y=y2) = 0.8. Hence, Investor II must be right - no matter what the value of Y is, the investment seems to be profitable.
The situation appears to be paradoxical. The information in posession of Investor I is independent of A and so useless. Despite that, he is able to confirm that the investment must appear profitable to the second investor. Hence, Investor I believes that the only rational choice for Investor II is to invest the money. What is good for Investor II must be good for Investor I. Thus, it appears that Investor I effectively found and confirmed an investment opportunity and, moreover he could have done this on his own, without receiving any advice from Investor II. This looks like a paradox because the information known to Investor I is independent of A and cannot possibly generate a better estmate of the probability of A than 0.5.
Question 3. Is the above example consistent with the laws of probability? It was argued in the last paragraph that the situation appears to be paradoxical. Perhaps the probability values and independence assumptions are inconsistent. (see the answer)
Question 4.
Answer Question 2 again. (see the answer)