Single-Leg Hop Distance per Launched Attempt: Landing Qualification and Exact Eight-Hop Calculations

Abstract

To determine how retaining failed landings in an eight-hop denominator changes the distance outcome, its variance, and the exposure needed to collect accepted hops. A launched hop contributed limb-length-normalized distance when it satisfied a two-second landing rule; an observed landing failure contributed zero qualified distance. Expectation, variance, and distance-exceedance identities were evaluated with two eight-outcome probability laws and 30 finite laws spanning five validity probabilities, three conditional means, and two distance shapes. Serial dependence, position-specific probabilities, and binary classification errors were evaluated by exact calculations. No participants were tested. Qualified distance equalled validity probability multiplied by mean valid-hop distance. Distribution A had seven valid outcomes with a conditional mean of 150% of limb length and a qualified mean of 131.25%; B had four valid outcomes with corresponding means of 180% and 90.00%. Collecting three accepted hops required 3.43 and 6.00 attempts on average under independent, stationary sampling. Lag-one correlation of 0.6 increased eight-hop mean variance by a factor of 3.0782. Opposing and aligned position trends yielded qualified means of 110.29 and 113.71 despite equal position-average components. Reclassifying A's three longest valid outcomes as failures reduced its mean to 73.25% and reversed the ranking against B. The launched-attempt denominator distinguishes distance among valid landings from qualified distance per hop. Interpretation depends on the acceptance rule, exposure count, and observation status.