Loading…
Reading the draws and working out what chance predicts.
LotteryOracle · F-49
HH and HT each have a one-in-four chance in two flips. Why does one take longer to arrive?
Probability in one specified window · 2 Flips: HH = HT = 25%
| Pattern (H = heads, T = tails) | Wins the race · Exact model | Observed / 2000 | Expected / 2000 |
|---|---|---|---|
| HH | 50% | 1013 | 1,000 |
| HT | 50% | 987 | 1,000 |
Flips · Expected: 3 · Observed: 3.018 · Runs stopped at 4,096 flips: 0
| Flips | HH | HT |
|---|---|---|
| 0 | 0 | 0 |
| 1 | 0 | 0 |
| 2 | 0.25 | 0.25 |
| 3 | 0.375 | 0.375 |
| 4 | 0.438 | 0.438 |
| 5 | 0.469 | 0.469 |
| 6 | 0.484 | 0.484 |
| 7 | 0.492 | 0.492 |
| 8 | 0.496 | 0.496 |
| 9 | 0.498 | 0.498 |
| 10 | 0.499 | 0.499 |
| 11 | 0.5 | 0.5 |
| 12 | 0.5 | 0.5 |
| 13 | 0.5 | 0.5 |
| 14 | 0.5 | 0.5 |
| 15 | 0.5 | 0.5 |
| 16 | 0.5 | 0.5 |
| 17 | 0.5 | 0.5 |
| 18 | 0.5 | 0.5 |
| 19 | 0.5 | 0.5 |
| 20 | 0.5 | 0.5 |
| 21 | 0.5 | 0.5 |
| 22 | 0.5 | 0.5 |
| 23 | 0.5 | 0.5 |
| 24 | 0.5 | 0.5 |
| 25 | 0.5 | 0.5 |
| 26 | 0.5 | 0.5 |
| 27 | 0.5 | 0.5 |
| 28 | 0.5 | 0.5 |
| 29 | 0.5 | 0.5 |
| 30 | 0.5 | 0.5 |
| 31 | 0.5 | 0.5 |
| 32 | 0.5 | 0.5 |
| 33 | 0.5 | 0.5 |
| 34 | 0.5 | 0.5 |
| 35 | 0.5 | 0.5 |
| 36 | 0.5 | 0.5 |
| 37 | 0.5 | 0.5 |
| 38 | 0.5 | 0.5 |
| 39 | 0.5 | 0.5 |
| 40 | 0.5 | 0.5 |
| 41 | 0.5 | 0.5 |
| 42 | 0.5 | 0.5 |
| 43 | 0.5 | 0.5 |
| 44 | 0.5 | 0.5 |
| 45 | 0.5 | 0.5 |
| 46 | 0.5 | 0.5 |
| 47 | 0.5 | 0.5 |
| 48 | 0.5 | 0.5 |
| 49 | 0.5 | 0.5 |
| 50 | 0.5 | 0.5 |
| 51 | 0.5 | 0.5 |
| 52 | 0.5 | 0.5 |
| 53 | 0.5 | 0.5 |
| 54 | 0.5 | 0.5 |
| 55 | 0.5 | 0.5 |
| 56 | 0.5 | 0.5 |
| 57 | 0.5 | 0.5 |
| 58 | 0.5 | 0.5 |
| 59 | 0.5 | 0.5 |
| 60 | 0.5 | 0.5 |
| 61 | 0.5 | 0.5 |
| 62 | 0.5 | 0.5 |
| 63 | 0.5 | 0.5 |
| 64 | 0.5 | 0.5 |
| 65 | 0.5 | 0.5 |
| 66 | 0.5 | 0.5 |
| 67 | 0.5 | 0.5 |
| 68 | 0.5 | 0.5 |
| 69 | 0.5 | 0.5 |
| 70 | 0.5 | 0.5 |
| 71 | 0.5 | 0.5 |
| 72 | 0.5 | 0.5 |
| 73 | 0.5 | 0.5 |
| 74 | 0.5 | 0.5 |
| 75 | 0.5 | 0.5 |
| 76 | 0.5 | 0.5 |
| 77 | 0.5 | 0.5 |
| 78 | 0.5 | 0.5 |
| 79 | 0.5 | 0.5 |
| 80 | 0.5 | 0.5 |
HH → HH
TTTTTHH → HH
TTTHH → HH
TTTTTTHT → HT
HH → HH
HT → HT
TTHT → HT
HT → HT
HT → HT
HH → HH
TTTTTTHT → HT
HH → HH
Fair independent coin flips; two distinct patterns of equal length, 2–5 symbols. An exact prefix-state Markov chain gives race odds and duration. Overlapping starts remain active. The curves are cumulative race-win probabilities, not single-pattern waiting distributions. The 2,000 seeded races stop at 4,096 flips if necessary; their average is capped when censoring occurs.
Fair independent coin flips; two distinct patterns of equal length, 2–5 symbols. An exact prefix-state Markov chain gives race odds and duration. Overlapping starts remain active. The curves are cumulative race-win probabilities, not single-pattern waiting distributions. The 2,000 seeded races stop at 4,096 flips if necessary; their average is capped when censoring occurs.
E[T(w)] = Σ{j : prefixⱼ(w)=suffixⱼ(w)} 2ʲ; (I−Q)u = b
After a failed attempt at HH, a tail erases the useful prefix. For HT, another head preserves a fresh start. Overlap changes the waiting time: HH needs six flips on average, HT four. This does not change the odds of any specified two flips.