Probability of trial sequence A trader learns to predict whether the stock price will rise or fall on a particular day of trading. To do this, he calls one hundred friends and asks them to toss a coin once a day, thus receiving one hundred signs of the type "heads / tails from the n-th friend." What is the probability that within a week of such analysis there will be a sign that correlates 100% with the dynamics of stocks?
Which distribution should I consider to solve that task? Bernoulli?
 A: Yes, Bernoulli distributions are involved.  But they are combined in a complex way; and it is important to be clear about how you understand the question.  Here is one proposal.
Framing the problem
Let us say that "100% correlates with the dynamics of stocks" means the following:

*

*Before conducting this experiment, you designate a set of frequently traded assets (the "market").


*Over the course of the seven days, you will compare the sum of closing prices of those assets to the same sum the previous day.  On day $d,$ set $Y(d)=1$ when the new sum is greater than the old one and otherwise let $Y(d)=0.$


*Let $Z(d,f)$ (which depends on the day $d$ and the friend $f$) equal $1$ when at the beginning of that day the friend flips a head and otherwise equal $0.$


*Assume every one of those coins is "fair:" this means the tosses are independent and, each time, have a $1/2$ chance of landing heads.
For each day $d$ and friend $f,$ let $X(d,f)=1$ when $Z(d,f)=Y(d):$ that is, your friend $f$'s flip agrees with (or "predicts") the direction in which the market moves on day $d$.
Assumption (4) implies all the $Z(d,f)$ are independent random variables and therefore the $X(d,f)$ are independent, too.  Moreover, every one of the 700 $X(d,f)$ variables have Bernoulli distributions with probabilities $1/2.$
The preceding observations are key, so make sure you understand them and can explain them to others.
Solution
To solve this problem, you now have to demonstrate:

*

*A friend $f$ achieves "100% correlation" when the sum of all the friend's $X(d,f)$ values (over all seven days) equals $7.$  This means their coin flips correctly "predicted" the movement of the market every day of the week.


*The sum of any particular friend's $X(d,f)$ has a Binomial$(7,1/2)$ distribution.


*Therefore, the chance that at least one friend achieves 100% correlation is the chance that the largest of $100$ independent draws from a Binomial$(7,1/2)$ distribution equals $7.$


*You can compute this chance as $1 - (1 - (1/2)^7)^{100} \approx 1 - \exp(-100/2^7) \approx 0.54.$
You may conclude that at the end of this experiment, it is more likely than not that at least one of your friends' coin flips will have predicted the market movements for a week.
Application
Before the experiment begins, ask your friends to document the times and results of their coin flips.  If one friend does achieve 100% correlation, send an email to a million gullible investors telling them about this friend only, along with that friend's documentation ("this market watcher predicted the market's movements correctly every day last week: you can have exclusive access to their next predictions!").  Offer advance information about this friend's future coin flips for a small price.  You could make a fortune.
Extra credit: explain the allegory.  Hint: a common term for friend is "market guru."

NB: Nothing in this post constitutes recommendation or advice.  Invest (and send emails) at your own risk, after consulting a lawyer about which of your country's laws you might be breaking if you carry out this program.
