Recipe

Quantify how an early streak gets diluted, not corrected

Use when: you're given a fixed imbalance (a streak or run) in the first few trials of a repeated random process, plus more trials assumed to split at the expected rate. You're asked how the running proportion or count behaves after that

Goal: the running proportion after the additional trials, plus confirmation that the streak's raw count imbalance is still exactly present, unchanged

Taught in Why probability doesn't "balance out" immediately

1. 1

   Compare the streak's actual count of the outcome to what a fair 50-50 split would give over that same number of trials.

   Example

   The first 8 flips of a fair coin are all tails. Expected tails in 8 flips is 8/2=4, so the excess is 8-4=4.

   • you're given a streak of raw outcomes (e.g. k heads in a row) — excess = actual streak count minus the expected count over that many trials.
   • you're given the imbalance already stated as a number (e.g. "ten more heads than expected") — use that number directly as the excess, and skip to Step 2.

   Why: this excess is the fixed, permanent quantity that later trials will dilute but never erase.

2. 2

   For the additional trials that follow the streak, assume they land at the expected 50-50 split unless stated otherwise. Add half of that count to the streak's own count to get the running total count of the outcome.

   Example

   20,000 more flips follow the streak above; their expected tails is 20{,}000/2=10{,}000. Running total tails: 8+10{,}000=10{,}008.

   Why: the additional trials contribute only their own expected share; they never retroactively touch what already happened during the streak.

3. 3

   Add the streak's own trial count to the additional trials' count to get the grand total, then divide the running total from Step 2 by that grand total.

   Example

   Grand total trials: 8+20{,}000=20{,}008. Running proportion: 10{,}008/20{,}008 \approx 0.50020.

   Why: the running proportion, not the raw count, is exactly what the law of large numbers describes converging toward the true probability.

4. 4

   Compute what the running total would be under an EXACT 50-50 split of the grand total, then subtract that from the actual running total found in Step 2. This difference should equal Step 1's excess exactly.

   Example

   Exact-split total: 0.5 \times 20{,}008 = 10{,}004. Actual running total: 10{,}008. Difference: 10{,}008-10{,}004=4, matching Step 1's excess of 4 exactly.

   Why: this is the distinction the whole section rests on. A shrinking PROPORTION alongside an unchanged raw COUNT difference is proof the mechanism is dilution, never correction.