Index Of: Luck By Chance

Title:

The Index of Luck by Chance: Quantifying Randomness in Outcomes and Perceptions of Serendipity

5. Applications & Interpretation

| Field | Luck index high (e.g., 0.8) means | Luck index low (e.g., 0.2) means | |--------|--------------------------------|--------------------------------| | Investing | Returns mostly random; indexing beats stock picking | Skill matters; active management may work | | Medicine | Most positive trial results false positives | Real treatment effects dominate | | Hiring | Who gets promoted is nearly random | Performance reviews reflect ability |

1. Introduction

Everyone has experienced luck—a unexpected win, a near miss, or an improbable failure. Yet, when analyzing performance (e.g., in sports, trading, or exams), we often conflate luck with skill. The Index of Luck by Chance seeks to formalize the proportion of an outcome’s deviation from expectation that is due purely to randomness. index of luck by chance

The ILC answers the question: Given a set of opportunities or trials, how likely is it that the observed success was simply a result of chance?

7. Practical Tools to Compute Your Own Luck Index

R – lme4 for variance components, boot for simulations.
Python – statsmodels for mixed models, numpy.random for Monte Carlo.
Excel – Use VAR.P, simulate with RAND() and NORM.INV(RAND(), ...).

2. Definition of the Index

Let:

( N ) = number of independent trials or opportunities.
( p ) = objective probability of success per trial (under a null model of pure chance).
( k ) = observed number of successes.

Under pure randomness, ( k ) follows a binomial distribution:
[ P(K = k) = \binomNk p^k (1-p)^N-k ]

The Index of Luck by Chance (ILC) is defined as the complement of the p-value for a one-tailed test of excess success (or failure), normalized to [0,1]: Title: The Index of Luck by Chance: Quantifying

For positive luck (more successes than expected by chance):
[ \textILC\textpos = 1 - \sumi=0^k-1 P(K=i) \quad \text(or capped at 1) ] Equivalently, ( \textILC_\textpos = P(K \geq k) ) — the probability that pure chance would produce at least as many successes.

For negative luck (fewer successes than expected):
[ \textILC_\textneg = P(K \leq k) ] Under pure randomness

Thus:

ILC = 0.5 means the outcome is exactly at the median of chance expectation.
ILC near 1 indicates a very lucky outcome (extreme deviation in favorable direction).
ILC near 0 indicates very bad luck (extreme unfavorable deviation).

A unified version:
[ \textILC = 2 \times \min\big(P(K \ge k), P(K \le k)\big) ] is sometimes used for deviation magnitude irrespective of direction.