** 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**)