Ask The Wizard #366
What is your recommended starting word in Wordle?
To answer that, I first looked at the frequency of each letter in each position, based on the list of allowed Wordle solutions.
Letter Frequency in Wordle
Letter | Pos. 1 | Pos. 2 | Pos. 3 | Pos. 4 | Pos. 5 | Total |
---|---|---|---|---|---|---|
A | 141 | 304 | 307 | 163 | 64 | 979 |
B | 173 | 16 | 57 | 24 | 11 | 281 |
C | 198 | 40 | 56 | 152 | 31 | 477 |
D | 111 | 20 | 75 | 69 | 118 | 393 |
E | 72 | 242 | 177 | 318 | 424 | 1233 |
F | 136 | 8 | 25 | 35 | 26 | 230 |
G | 115 | 12 | 67 | 76 | 41 | 311 |
H | 69 | 144 | 9 | 28 | 139 | 389 |
I | 34 | 202 | 266 | 158 | 11 | 671 |
J | 20 | 2 | 3 | 2 | 0 | 27 |
K | 20 | 10 | 12 | 55 | 113 | 210 |
L | 88 | 201 | 112 | 162 | 156 | 719 |
M | 107 | 38 | 61 | 68 | 42 | 316 |
N | 37 | 87 | 139 | 182 | 130 | 575 |
O | 41 | 279 | 244 | 132 | 58 | 754 |
P | 142 | 61 | 58 | 50 | 56 | 367 |
Q | 23 | 5 | 1 | 0 | 0 | 29 |
R | 105 | 267 | 163 | 152 | 212 | 899 |
S | 366 | 16 | 80 | 171 | 36 | 669 |
T | 149 | 77 | 111 | 139 | 253 | 729 |
U | 33 | 186 | 165 | 82 | 1 | 467 |
V | 43 | 15 | 49 | 46 | 0 | 153 |
W | 83 | 44 | 26 | 25 | 17 | 195 |
X | 0 | 14 | 12 | 3 | 8 | 37 |
Y | 6 | 23 | 29 | 3 | 364 | 425 |
Z | 3 | 2 | 11 | 20 | 4 | 40 |
Then I looked at all the words in the Wordle solution list with five distinct letters and scored them according to the letter frequency table above. I awarded two points for a match in the correct position and one point for a match in an incorrect position. Then I sorted the list, which you see below.
Best Starting Words in Wordle
Rank | Word | Points |
---|---|---|
1 | Stare | 5835 |
2 | Arose | 5781 |
3 | Slate | 5766 |
4 | Raise | 5721 |
5 | Arise | 5720 |
6 | Saner | 5694 |
7 | Snare | 5691 |
8 | Irate | 5682 |
9 | Stale | 5665 |
10 | Crate | 5652 |
11 | Trace | 5616 |
12 | Later | 5592 |
13 | Share | 5562 |
14 | Store | 5547 |
15 | Scare | 5546 |
16 | Alter | 5542 |
17 | Crane | 5541 |
18 | Alert | 5483 |
19 | Teary | 5479 |
20 | Saute | 5475 |
21 | Cater | 5460 |
22 | Spare | 5457 |
23 | Alone | 5452 |
24 | Trade | 5449 |
25 | Snore | 5403 |
26 | Grate | 5403 |
27 | Shale | 5392 |
28 | Least | 5390 |
29 | Stole | 5377 |
30 | Scale | 5376 |
31 | React | 5376 |
32 | Blare | 5368 |
33 | Parse | 5351 |
34 | Glare | 5340 |
35 | Atone | 5338 |
36 | Learn | 5324 |
37 | Early | 5320 |
38 | Leant | 5307 |
39 | Paler | 5285 |
40 | Flare | 5280 |
41 | Aisle | 5280 |
42 | Shore | 5274 |
43 | Steal | 5268 |
44 | Trice | 5267 |
45 | Score | 5258 |
46 | Clear | 5258 |
47 | Crone | 5253 |
48 | Stone | 5253 |
49 | Heart | 5252 |
50 | Loser | 5251 |
51 | Taper | 5248 |
52 | Hater | 5243 |
53 | Relay | 5241 |
54 | Plate | 5240 |
55 | Adore | 5239 |
56 | Sauce | 5236 |
57 | Safer | 5235 |
58 | Alien | 5233 |
59 | Caste | 5232 |
60 | Shear | 5231 |
61 | Baler | 5230 |
62 | Siren | 5226 |
63 | Canoe | 5215 |
64 | Shire | 5213 |
65 | Renal | 5210 |
66 | Layer | 5206 |
67 | Tamer | 5200 |
68 | Large | 5196 |
69 | Pearl | 5196 |
70 | Route | 5194 |
71 | Brace | 5192 |
72 | Slice | 5178 |
73 | Stage | 5171 |
74 | Prose | 5170 |
75 | Spore | 5169 |
76 | Rouse | 5166 |
77 | Grace | 5164 |
78 | Solar | 5152 |
79 | Suite | 5150 |
80 | Roast | 5145 |
81 | Lager | 5130 |
82 | Plane | 5129 |
83 | Cleat | 5129 |
84 | Dealt | 5128 |
85 | Spear | 5126 |
86 | Great | 5126 |
87 | Aider | 5123 |
88 | Trope | 5116 |
89 | Spire | 5108 |
90 | Tread | 5107 |
91 | Slave | 5097 |
92 | Close | 5090 |
93 | Lance | 5090 |
94 | Rinse | 5088 |
95 | Cause | 5087 |
96 | Prone | 5087 |
97 | Drone | 5082 |
98 | Noise | 5079 |
99 | Crest | 5073 |
100 | Sober | 5068 |
So, there you have it, my recommended starting word, which I use, is STARE.
What is i^i
Suppose a casino has a game based on a fair coin flip that pays even money. A player wishes to play one million times at $1 a bet. How much money should he bring to the table to have a 50% chance of not going broke?
Let's first answer the question of what is the probability the player will be down more than x units after one million flips, assuming the player has an unlimited bankroll.
Since this is a fair bet, the mean win after a million flips is zero. The variance of each flip is 1, so the variance of one million flips is one million. One standard deviation is thus sqrt(1,000,000) = 1000.
We can find the bankroll required with the Excel function =norm.inv(probability,mean,standard deviation). For example, if we put in =norm.inv(.25,0,1000), we get -674.49. This means if after one million flips, the player has a 25% chance of being down 674 or more. Please keep in mind this is an estimate. To get a true answer, we should use the binomial distribution, which would be very tedious with a million flips.
It could very well happen that if the player took $674 to the table, he might run out of money before the million flips. If he could keep playing on credit, it might happen that he has a recovery and finishes less than $674 down. In fact, once the player is at -674, there is a 50/50 chance he will end up above or below -674 at any given point in the future.
So, if the player can play on credit, there are three possible outcomes.
- Player never falls below -674.
- Player falls below -674 at some point, but recovers and finishes above -674.
- Player falls below -674 at some point, keeps playing and loses even more.
We have established scenario 3 has a probability of 25%.
Scenario 2 must have the same probability as scenario 3, because once the player is down -674, he has a 50/50 chance to finish above or below that point after one million flips.
Scenario 1 is the only other alternative, which must have probability 100%-25%-25% = 50%.
If the probability the player never falls below 674 is 50%, then the alternative of falling below must be 100%-50% = 50%.
So, there is our answer to the original question, $674.
This question is asked and discussed in my forum at Wizard of Vegas.
You wish to play a game that requires two ordinary six-sided dice. Unfortunately, you lost the dice. However, you have nine index cards, which you may mark any way you like. The player must choose two index cards randomly from the nine, without replacement, and take the sum of the two cards.
Mark the cards as follows:
1 @ 0.5
1 @ 1.5
2 @ 2.5
1 @ 3.5
2 @ 4.5
1 @ 5.5
1 @ 6.5
This question is asked and discussed in my forum at Wizard of Vegas.