Problem: di birthday wheel
Ow much spin bifo some numba come again? Far less dan most people guess.
Di birthday problem pon a 37-pocketgearPocketOne a di 37 compartment wid numba pon di rotor. Nex name: slot, cup.Read di whole ting → wheel: afta jus 8 spin, chance betta dan even seh some numba done show up two time a ready. Playa read repeat as pattern; dem a jus wah a fair wheel fi do.
Di problem
Spin a fair European wheel. Ow much spin till di probabilitymathsProbabilityDi chance fi a result, fram 0 to 1 or 0% to 100%. One numba: 1/37 = 2.70%.Read di whole ting → seh at least two spin show di same numba pass one half? Most playa guess somewhere roun 18 or 19, half a 37. Di ansa a 8.
Di solution
Work out di probability fi nuh repeat an tek it fram one. Afta k spin, p(all different) = (37/37)(36/37)(35/37)…((38 − k)/37). Fi k = 8: 1 × 0.973 × 0.946 × 0.919 × 0.892 × 0.865 × 0.838 × 0.811 ≈ 0.44, suh p(some repeat) ≈ 0.56. Fi k = 14 it a 0.93; fi k = 20 it a 0.996. Yuh gut feelin fail cause di numba a pair a spin grow like k², an any pair can match.
Wah mek it matta at di table
A result board a di las 20 numba wi nearly always show at least one repeat, often two or tree. Playa see “17 come two time inna ten spin” an decide seh 17 hot; di birthday arithmetic seh a repeat inside ten spin happen 74% a di time an di numba weh repeat a random. Di pattern real; di meanin nuh real.
Di follow-up problem
Ow much spin till one specific numba, seh 17, show up two time? Dat a different question wid a much bigga ansa: di expected wait a 74 spin, an di probability fi get two 17 inside 20 spin a bout 10%. Di difference between “some numba repeat” (8 spin) an “dis numba repeat” (74 spin) a di whole difference between a pattern yuh notice afta an a prediction yuh mek bifo.
Try it fi yuhself
Run di Demo Game fi ten spin an look pon di history. Usually a repeat wi deh deh. Run it again. Di numba weh repeat wi different. Do dis ten time an yuh see di birthday problemmathsBirthday problemOw much people it tek bifo two a dem probably share a birthday, or ow much spin bifo a numba probably come again. Pon a 37-pocket wheel, 8 spin enough fi a betta-dan-even chance.Read di whole ting →; do it a tousand time (di simulator dem can do it) an yuh measure it.
Formula
Question dem
Exactly, wid 37 pocket instead a 365 day. 23 people give 50% fi birthday; 8 spin give 56% fi di wheel.
50% afta bout 19 spin. Still a regular ting.
No. Tousands a spin fram one wheel coulda show dat; a repeat neva do.
Fun wid maths
Try dem yuhself fus. Evry ansa fold up; open it wen yuh ready.
A result board beside a American wheel show di las 16 spin. Wah di chance seh nuh numba show up two time pon it?
Show di ansa
Ansa Bout 2.5%: roughly 39 board inna 40 show at least one repeat.
Go spin by spin. Di fus spin can be anyting. Di second haffi miss it: 37 chance inna 38. Di third haffi miss both: 36 inna 38. Di sixteenth haffi miss fifteen numba: 23 inna 38.
Multiply: 38/38 × 37/38 × 36/38 × … × 23/38 ≈ 0.0246.
A board wid nuh repeat a di rare one. A board wid a repeat nuh seh nutten bout di wheel.
Inna ten spin a a European wheel, wah di chance seh 17 come up exactly five time? Seh sum numba, any numba, come up five or more time? An seh sum numba come up at least tree time inna four spin?
Show di ansa
Ansa 17 exactly five time: 1 inna 315,576. Sum numba five or more time: bout 1 inna 8,333. Sum numba tree or more time inna four spin: 1 inna 349.
17 five time: pick which five a di ten spin, 252 way, each one wid chance (1/37)⁵ × (36/37)⁵. Togedda, 1 inna 315,576.
Sum numba: each a di 37 numba have a chance a 1 inna 308,325 fi five or more hit, weh gi 37 inna 308,325, or 1 inna 8,333. Di rare case a two numba wid five hit each get count two time, but fi tek it out nuh change nutten at dis precision.
Tree time inna four spin: fi one numba, 4 × 36 + 1 = 145 a di 37⁴ result weh possible; fi any a di 37 numba, 145/37³ = 1 inna 349.
Fi a double-zeropon di wheelZeroDi one green pocket weh no even-money, dozen or column bet nuh cover; a weh di house edge come fram. Nex name: single zero, 0.Read di whole ting → wheel, di matchin figure dem a 1 inna 359,275, 1 inna 9,243 an 1 inna 368. Di gap between “17” an “sum numba” a di birthday problemmathsBirthday problemOw much people it tek bifo two a dem probably share a birthday, or ow much spin bifo a numba probably come again. Pon a 37-pocket wheel, 8 spin enough fi a betta-dan-even chance.Read di whole ting → again: it have 37 way fi get surprise.
Sumting fi laugh
Sumting repeat early. Dat sumting a random.
Source dem
- Feller, W. An Introduction to Probability Theory and Its Applications, Vol. 1.
- Wizard of Odds. “Ask the Wizard”, roulette questions.