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Ask the Wizard #374

You shuffle a standard (28 tile) set of dominos and draw one at random. You carefully uncover one of the sides to reveal a 6. What is the probability that this is the double 6 tile?

Gialmere

For the benefit of those not familiar with dominoes, a domino has two sides, each with a number from 0 to 6. A set consists of one of each possible domino. That would be all combin(7,2)=21 ways to choose numbers, without replacement, out of 7, plus all 7 ways to have both numbers on the same side.

Here are all all 28 of them: 0-0, 0-1, 0-2, 0-3, 0-4, 0-5, 0-6, 1-1, 1-2, 1-3, 1-4, 1-5, 1-6, 2-2-2, 2-3, 2-4, 2-5, 2-6, 3-3, 3-4, 3-5, 3-6, 4-4, 4-5, 4-6, 5-5, 5-6, 6-6.

With that explanation out of the way, here are the answer and solution.

1/4 = 25%

A simple way to reword the question is what is the probability a randomly chosen domino has the same number on both sides. That answer is simply 7/28 = 1/4.

You might say, the answer should be 1/7, because the other side could be any of the seven possible numbers. It's true it could be any of the seven numbers, but they are not all equally likely. The 6-6 domino should be double counted, because it has two sides with a six. So, the other side could be either of two sixes and there are eight total sixes in the set. Thus the answer is 2/8 = 1/4.

This question is asked and discussed in my forum at Wizard of Vegas. The question is dedicated to the memory of Alan Mendelson.

Alan and Bob decide to play a tennis until one of them wins two games in a row. The probability of Alan winning any given game is 2/3. What is the probability Alan wins the tournament?

"Anonymous" .

Alan and Bob decide to play a tennis until one of them wins two games in a row. The probability of Alan winning any given game is 2/3. What is the probability Alan wins the tournament?

The answer is 16/21 = 76.1905%.

 

The following is the generation solution for any probability of Alan winning an individual game.

Let p = probability a wins.
Let a = probability Alan wins tournament after Alan wins last game.
Let b = probability Alan wins tournament after Bob wins last game.

(1) a = p + (1-p)*b
(2) b = p*a

Substituting equation (2) into equation (1):

a = p + (1-p)*pa
a = p + pa - p2*a
a - pa + p2*a = p
a(1-p+p2) = p
(3) a = p/(1-p+p2)

Substituting equation (3) into equation (2):

b = p2/(1-p+p2)

The first game will determine whether the probability of a winning the tournament is a or b:

answer = pa + (1-p)b
= p2/(1-p+p2) + (1-p)*p2/(1-p+p2)
= (2a2-a3)/(a2-a+1)

Plugging a=2/3, gives a probability of winning the tournament of 16/21.

This question is asked and discussed in my forum at Wizard of Vegas.

In Ask the Wizard #366 you stated your recommended starting word in Wordle is STARE. My question is what is your recommended second word, given that you started with STARE, according to the outcome of turn 1?

"Anonymous" .

In theory, there are 35 = 243 possible outcomes of the first turn. Forgive me if I only address those where the sum of yellows and greens is 0 to 2. I'll break up my advice into categories of how many yellows and greens the player received on the first turn. I will indicate the ten best allowable words, according to my analysis, as well as my personal pick, which takes into consideration the placement on the list and my opinion of the likelihood the New York Times will choose that word. In some cases, there were no words that fit a particular score. In those cases, I put an "X" for that line.

0 Yellows — 0 Greens

Turn 1 Score Wizard Pick Top 10
----- DOILY BLINY,DOILY,PIONY,NOILY,FONLY,PHONY,PONCY,BLIMY,WHINY,CONKY

0 Yellows — 1 Green

Turn 1 Score Wizard Pick Top 10
----G CLONE CLINE,CLONE,CLONE,MOILE,CHINE,BOULE,CHILE,VOILE,BONIE,POULE
---G- GLORY FLORY,GLORY,LOURY,COWRY,DOWRY,CHIRL,COURD,LOWRY,BOURN,BOURD
--G-- FOAMY COALY,COADY,BIALY,FOAMY,BLADY,LOAMY,HOAGY,FLAMY,FLAKY,GLADY
-G--- ITCHY ITCHY
G---- SHINY SOILY,SHINY,SONLY,SHILY,SLIMY,SPINY,SLOPY,SOUPY,SPICY,SHOWY

1 Yellow — 0 Greens

Turn 1 Score Wizard Pick Top 10
----Y MONEY CONEY,BONEY,COLEY,CLUEY,PONEY,BLUEY,POLEY,FOLEY,COVEY,MONEY
---Y- PRICY CRONY,BRINY,PRICY,CORNY,ROILY,DRILY,DRONY,PORNY,PRIMY,CURNY
--Y-- DAILY GAILY,DAILY,HAILY,GAUCY,FANCY,BADLY,MANLY,CANDY,DANCY,COPAY
-Y--- POINT POINT,CLINT,POUTY,COUNT,DOILT,PONTY,GOUTY,BIONT,BUILT,BUILT
Y---- NOISY BOUSY,NOISY,MOUSY,COPSY,LOUSY,DONSY,MOPSY,COINS,CUISH,BOSKY

0 Yellows — 2 Greens

Turn 1 Score Wizard Pick Top 10
---GG CHORE MOIRE,COURE,BLORE,CHORE,LOURE,QUIRE,WHORE,KIORE,POWRE,FIBRE
--G-G PLANE PLANE,FLANE,DIANE,PLACE,POAKE,LIANE,CLADE,GLACE,CLAME,CLAVE
--GG- DIARY CLARY,GOARY,HOARY,CHARY,FLARY,GLARY,DIARY,BOARD,COARB,UNARY
-G--G UTILE UTILE
-G-G- X X
-GG-- OTAKU OTAKU
G---G SLICE SLICE,SOUCE,SHINE,SHONE,SONCE,SLIDE,SLIME,SLIPE,SLIVE,SPINE
G--G- SHIRK SPIRY,SHORL,SHIRK,SHORN,SLORM,SWIRL,SKIRL,SLURP,SPORK,SWORN
G-G-- SOAPY SOAPY,SOAPY,SHALY,SHADY,SPACY,SWALY,SCALY,SHAKY,SNAKY,SNAKY
GG--- STIMY STONY,STIMY,STOGY,STIVY,STUDY,STUDY,STINK,STINK,STICH,STOND

2 Yellows — 0 Greens

Turn 1 Score Wizard Pick Top 10
---YY CRUEL COREY,GOIER,CRIED,PLIER,FLIER,ROPEY,BONER,PRIED,FOUER,FRIED
--Y-Y CAGEY CAGEY,WANEY,CAKEY,PACEY,GAMEY,FAKEY,LACEY,WAVEY,MAZEY,CANEH
--YY- RAINY RAINY,CARNY,BARNY,PARLY,FARCY,FORAY,FORAY,MARLY,CARDY,DARCY
-Y--Y DEITY TONEY,MOTEY,CUTEY,FOUET,DEITY,BLUET,COMET,COVET,FEINT,GOETY
-Y-Y- PRINT BORTY,PRINT,TRULY,TRULY,TRIPY,PORTY,FRONT,FORTY,DORTY,CROUT
-YY-- PAINT LAITY,PAINT,CANTY,FAINT,FAINT,BANTY,DAINT,PANTY,PATLY,TALCY
Y---Y COSEY COSEY,POSEY,MOSEY,POESY,HOSEY,NOSEY,CLIES,COSED,CONES,PLIES
Y--Y- BRISK BROSY,PROSY,GRISY,CRUSY,GORSY,DRUSY,BIRSY,HORSY,PURSY,BRISK
Y-Y-- DAISY DAISY,PANSY,MAUSY,PALSY,CASKY,CAINS,GAWSY,BAILS,PAINS,FAINS
YY--- FOIST TOUSY,FOIST,FOIST,MOIST,POTSY,BLIST,BUIST,HOIST,HOIST,BITSY

1 Yellows — 1 Green

Turn 1 Score Wizard Pick Top 10
GY--- SILTY SUINT,SOUCT,SILTY,SOFTY,SLUIT,SLIPT,SHUNT,SPILT,SHOUT,SLOTH
G-Y-- SAUCY SAUCY,SADLY,SANDY,SAVOY,SAICK,SAUCH,SLOAN,SHOAL,SPLAY,SOLAH
G--Y- SURLY SURLY,SURGY,SURFY,SHIUR,SUDOR,SOHUR,SCOUR,SIROC,SIRUP,SPROD
G---Y SHLEP SKIEY,SHIEL,SHIED,SOLED,SHOED,SLUED,SPIEL,SEPOY,SENVY,SINED
YG--- X X
Y-G-- CLASH CLASH,BLASH,COALS,PLASH,FLASH,CLASP,FLASK,CLANS,FOALS,MOANS
Y--G- YOURS COIRS,COURS,LOIRS,FLIRS,POURS,NOIRS,FLORS,FOURS,MUIRS,BLURS
Y---G POISE POISE,POISE,CLOSE,BOUSE,HOISE,CHOSE,NOISE,GUISE,DOUSE,MOUSE
-GY-- ATOPY ATIMY,ATOMY,ATOPY
-G-Y- X X
-G--Y ETHIC ETHIC,ETHYL
-YG-- PLANT GOATY,TOADY,PLATY,COAPT,PLANT,CHANT,PLAIT,GIANT,CLAUT,DIACT
-Y-G- COURT COURT,CHIRT,FLIRT,BLURT,QUIRT,WHORT,THIRL,NITRY,YOURT,THIRD
-Y--G THINE BOITE,TOILE,CLOTE,BLITE,TUILE,CONTE,FLITE,THINE,FLOTE,THOLE
--GY- CRAZY CRAPY,BRAKY,CRAZY,ORACY,BRAXY,GRAPY,GRAVY,CRANK,BRAIL,BRAND
--G-Y LEAKY BEANY,MEANY,LEANY,MEALY,BEADY,BEAMY,BEAKY,VEALY,PEAKY,PEAVY
--YG- FAIRY FAIRY,DAIRY,DAIRY,LAIRY,HAIRY,VAIRY,CAIRN,CAIRN,CAIRD,BAIRN
--Y-G CANOE FAINE,DAINE,MAILE,ALINE,HAOLE,CANOE,ALONE,ALONE,BAIZE,PANCE
---GY DECRY FIERY,CEORL,HENRY,DECRY,QUERY,QUERY,WEIRD,CLERK,CLERK,ONERY
---YG BRINE CRINE,CRONE,BRINE,PRONE,PRICE,PROLE,GRONE,DRONE,DRONE,GRICE

An argument could be made that with one green one should do a probing turn with five fresh letters. If you do that, I would go with DOILY. To be honest, I am not sure if you should go for the possible score of 2 or play it safe and try a fifth letter.