Luck Index

Luck Index definition and calculation

Luck Index Definition

The Luck Index (LI) of an individual is a quantitative measure ranging from 0 to 1, designed to represent the proportion of positive outcomes achieved by the individual in a specified set of trials, where the outcomes are presumed to be influenced by variables beyond the individual's control and skill. The index is calculated based on the ratio of successful outcomes (wins) to the total number of trials, adjusted for the expected probability of success in those trials to account for varying degrees of difficulty or chance.

Mathematical Formulation:

$$LI= \frac{1}{N}∑^N_{i=1} \frac{W_i−E(W_i)} {1−E(W_i)}$$

Where:

• $N$ is the total number of trials or opportunities.
• $W_i$ is a binary indicator of success in $i^{th}$ trial (1 if the trial was a win, 0 otherwise).
• $E(W_i​)$ is the expected probability of winning the trial $i^{th}$, based on an objective assessment of the chance involved.

This formulation attempts to normalize the success rate by the expected success rate, allowing for a fair comparison across different contexts and ensuring that the Luck Index is not merely a reflection of the difficulty of the challenges faced.

Adjustments and Considerations:

1. Normalizing for Expected Probability: The subtraction of $E(W_i​)$ from $W_i​$ and the division by $1−E(W_i​)$ normalize for the expected probability of winning, making the index more reflective of luck rather than skill or difficulty level.
2. Handling Varied Probabilities: By incorporating $E(W_i​)$, the index accounts for situations with different inherent probabilities of success. For instance, a win in a highly unpredictable game would be weighted differently from a win in a more predictable scenario.
3. Range and Interpretation: The Luck Index is constrained between 0 and 1, where 0 indicates an absence of luck (never winning beyond what is expected by chance alone) and 1 indicates perfect luck (always winning more often than expected by chance alone).

4. Validity Across Domains: The definition assumes that the set of trials or opportunities is sufficiently large and diverse to validly represent the concept of luck across various contexts in which luck might play a role.

This scientific definition aims to quantify luck in a way that is independent of the specific domain, relying on statistical principles to provide a standardized measure of an individual's luck as observed through outcomes presumed to be influenced significantly by chance.