## Probability of luck

Let's define luck in the following way. Given a series of events $E = \{E_{1}, \dots, E_{m}\}$, where each event $E_{t}$ has a different outcome $e_{ti}$ with $i = 1, \dots, n$ for each person in a set of people $P$ with $n = |P|$, luck $L$ for a person $p \in P$ is given by $$L_{p} = \mu_{p} - \frac{1}{n} \sum_{i=1}^{n} \mu_{i}$$ Where $\mu_{i} = \frac{1}{m} \sum_{t=1}^{m} e_{ti}$.

That is, luck for a person is the difference between the mean of the outcomes that person gets in the events considered and the mean of the outcomes of the group. We'll assume outcomes are independent from each other.

Events can be seen as a matrix of independent and identically distributed random variables of cardinality $m \times n$, each variable with mean $\mu$ and variance $\sigma^{2}$. My question is, given this definition, what can we say about luck?

One thing we can say is that it's highly improbable no one will have positive luck. That's because each column of our matrix $X_{1},\dots,X_{m}$ is a sample taken from the probability distribution of $X$. The probability of luck for a person $p$ being different from zero is \begin{align} P(L_{p} != 0) = \\[2mm] 1 - P(L_{p} = 0) = \\[2mm] 1 - P\left(\mu_{p} = \frac{1}{n} \sum_{i=1}^{n} \mu_{i}\right) \end{align}

As $n$ gets bigger, for the central limit theorem, $\frac{1}{n} \sum_{i=1}^{n} \mu_{i}$ can be approximated by a normal distribution $N(\mu, \frac{\sigma}{\sqrt{n} \sqrt{m}})$. The same can be said about $\mu_{p}$, approximated by $N(\mu, \frac{\sigma}{\sqrt{m}})$.

That means that $\mu_{p} - \frac{1}{n} \sum_{i=1}^{n} \mu_{i}$ is distributed according to a normal distribution of mean $\mu - \mu = 0$ and standard deviation $\frac{\sigma}{\sqrt{m}} - \frac{\sigma}{\sqrt{n}\cdot\sqrt{m}}$, let's call it $Y$.

Then: \begin{align} 1 - P(Y = 0) =\\[2mm] 1 - \frac{1}{\sigma_{Y} \sqrt{2\pi}} e^{-\frac{1}{2}\left( \frac{0-\mu_{Y}}{\sigma_{Y}} \right)} =\\[2mm] 1 - \frac{1}{\sigma_{Y} \sqrt{2\pi}} \end{align}

Which isn't zero. This actually includes the probability of being unlucky, so we have to halve it.

If we consider as a sample not only the series of successes of a person but also a set of people successes at each event time t, also this new samples means are normally distributed, with calculations roughly as above.

So the probability of a sample (an event for all the people) changing how luck is distributed among people is nonzero.

Can we say that it's unlikely for luck to stay the same? That's the case of all ones or all zeros in our sample, that is being in the extreme tails of the distribution.

Otherwise the wheel of fortune turns, as normal.