The underlying probability space

We will show that the example is consistent with the laws of probability. We will present a probability space on which one can define an event A, random variables X and Y and numbers x1, y1 and y2 with the following properties.

1. P(A) = 0.5
2. X is independent of A
3. P(Y=y1 | X=x1) + P(Y=y2 | X=x1) = 1
4. P(A | Y=y1) = 0.7
5. P(A | Y=y2) = 0.8
Consider i.i.d. random variables Z(j), for j = 1,2,...,10, such that P(Z(j) = 1) = P(Z(j) = 0) = 0.5. Let
• A = {Z(10) = 1},
• X be the sum of Z(j) from j = 1 to j = 9,
• Y be the sum of Z(j) from j = 1 to j = 10,
• x1 = 7
• y1 = 7
• y2 = 8
It is elementary to check that the properties (1)-(5) are satisfied.

One may wonder if the "paradox" is a result of choosing a very unrealistic scenario. In other words, is the probability that X takes the value x1 or larger exceedingly small? It is equal to about 9%, which seems to be quite reasonable.

Answer to Questions 1, 2 and 4.

The information obtained from Investor II by Investor I does not change his assessment of the probability of A. Note that the event {X=x1 and P(A|Y) > 0.6} is equal to {X=x1} so

P(A | X=x1 and P(A|Y) > 0.6) = P(A | X=x1) = P(A) = 0.5.

Investor I should not participate in the investment.

Conclusion.

It turns out that it is logically consistent and fully rational for

• Investor II to consider the investment profitable and
• Investor I to disregard the second investor's assessment of the probability of A despite his own lack of knowledge of any event or random variable that is not independent from A.
This appears paradoxical in view of our everyday experience. Consider any investment opportunity depending on the outcome of some random action A. We all have access to great volumes of "insider" information (i.e., information which is not available to the public) and which is independent from A. For example, you know what you had for breakfast today, which books you read in the last two weeks, etc. We instinctively disregard all the information that falls into the "insider and independent" category as totally useless. In such circumstances, a friendly advice based on some information which is not independent of A is normally regarded as worthy of following. The "paradox" presented on this page shows that various pieces of information can interplay in a way that may contradict the intuition.