Luck

Abstract

Luck is a matter of the positive (or negative) outcome for someone of a contingently chancy development that can yield positivities or negativities for this individual. One is lucky when, in a situation of uncertain outcome, one fares better than one has a right to expect, and unlucky when one fares worse. And this matter of an outcome-antecedent “right to expect” is inherently probabilistic in nature.

Appendix 1: Measuring Luck

• General Formulas

Odds-luck index::

1−p⁺ (where p⁺ is the (prior) probability of success/winning)

Yield-luck measure::

$\uplambda = Y{-}E$ (λ for amount of luck)

Max-luck::

$\lambda^{ + } = Y^{ + } {-}E$ (Y⁺ being the yield of the best possible outcome)

Proportionate luck::

$\uplambda^{\text{ * }} =\uplambda/\uplambda^{ + } = (Y{-}E)/(Y^{ + } - E)$

• Binary Gambles

Expectation::

$E = p \cdot W + (1 - p) \cdot L$

Stake::

$\Delta = W - L$

Win-luck::

$\uplambda({\text{win}}\} = (1 - p) \cdot \Delta = (1 - p)\,(W - L)$

• Multi-Outcome Situations

Expectations::

$E = \sum\limits_{k} {(p_{k} \cdot {<}{O_{k}}{>})}$ (where p_k is the prior probability of outcome O_k)

Yield luck:

$$\lambda \{ O_{i} \} = {<}{O_{i}}{>} - E$$

Notes::

• (1) Y is the outcome yield and E the expected yield.

• (2) <O_i> is the yield of outcome O_i.

• (3) With binary gambles, W is the win-yield and L the loss-yield.

• (4) p is the (prior) probability of the outcome at issue. With success-luck in binary gambles it will be the probability of winning.