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Ask The Wizard #374
Imagine you have a standard set of dominos with 28 tiles. You randomly select one and reveal one side that shows a 6. What are the odds that this tile is the double six?
To clarify for those unfamiliar with dominos, each domino has two faces, numbered from 0 to 6. A complete set comprises every unique domino. This includes all combinations of choosing 2 numbers from 7, which amounts to combin(7,2)=21, in addition to the 7 pairs where both sides share the same number.
Here’s the complete list of all 28 dominos: 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, 2-3, 2-4, 2-5, 2-6, 3-3, 3-4, 3-5, 3-6, 4-4, 4-5, 4-6, 5-5, 5-6, and 6-6.
Having explained that, let's proceed to the answer and its reasoning.
To put the question in simpler terms: what are the chances that a domino drawn at random has identical numbers on both ends? The result is straightforward: 7 out of 28, which simplifies to 1/4.
You may argue that the answer should be 1/7 since the other side could potentially display any of the seven numbers. While it’s true, the probabilities aren’t equal. The double 6 needs special consideration, as it counts for both sides when considering the total number of sixes. Hence, we have two sides showing a six out of a total of eight sixes in the set. This leads to the conclusion that the probability is 2 out of 8, simplifying to 1/4.
This topic has been brought up and analyzed in my online forum at Wizard of Vegas . This discussion is dedicated to the memory of Alan Mendelson.
Alan and Bob have decided to engage in a tennis match, with the objective that one of them must win two consecutive games to claim victory. Given that Alan has a 2/3 chance of winning each game, what is the probability that he will win the overall match?
Alan and Bob have decided to engage in a tennis match, with the objective that one of them must win two consecutive games to claim victory. Given that Alan has a 2/3 chance of winning each game, what is the probability that he will win the overall match?
The solution below outlines how to determine the probability of Alan winning a single game.
Let p = probability a wins.
Let 'a' signify the probability of Alan winning the tournament assuming he takes the last game.
Let 'b' signify the probability of Alan winning the tournament if Bob secures the last game.
(1) a = p + (1-p)*b
(2) b = p*a
Now, by 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)
Next, we substitute equation (3) into equation (2):
b = p2/(1-p+p2)
The outcome of the first game will dictate whether the probability for Alan's tournament win is represented by 'a' or 'b':
answer = pa + (1-p)b
= p2/(1-p+p2) + (1-p)*p2/(1-p+p2)
= (2a2-a3)/(a2-a+1)
Using a = 2/3 yields a total probability of Alan winning the tournament as 16/21.
This topic has been brought up and analyzed in my online forum at Wizard of Vegas .
In Ask the Wizard #366 You mentioned that for your Wordle strategy, your initial guess is 'STARE'. I’m curious about what you would suggest for your second guess based on the result of your first attempt.
In theory, there are 35= 243 potential outcomes can arise from the first guess. Allow me to focus specifically on scenarios where the combined count of yellow and green letters ranges between 0 and 2. I will categorize my suggestions based on how many yellows and greens were revealed in the initial guess. I will present a list of ten optimal words based on my analysis, along with my personal recommendation reflecting its position on the list and the likelihood of it being selected by the New York Times. If no fitting words were found for a certain scoring outcome, I have placed an 'X' for that entry.
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, SHALY, SHADY, SPACY, SWALY, SCALY, SHAKY, SNAKY, 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- | 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 |
One could argue that if a player has one green, it would be wise to select a test word featuring five new letters. If that’s the case, I recommend using DOILY. Honestly, I’m uncertain whether to aim for a potential score of 2 or to play it safe and try for an additional fifth letter.