I’d Like to Buy a Vowel

November 26, 2012 1 comment

The game show Wheel of Fortune has been a television staple for more than thirty years.  A hybridization of Hangman and Roulette, the game requires a way with words and a little bit of luck in order to succeed.

In case it’s been a while since you invited Pat Sajak and Vanna White into your home, here’s a quick rundown of the rules: contestants are presented with a blank word puzzle, and guess letters one at a time to gradually reveal a hidden message. Players earn money for each consonant they guess correctly; the amount they earn is determined by a spin of the wheel.  To guess a vowel, however, contestants must pay \$250.

Does this make sense, though?  Should a U cost as much as an E?  Certainly E shows up more often, so guessing an E may get you more bang for your buck.  On the other hand, since U is a rarer vowel, its appearance in a puzzle may give you more information than a couple of E’s would.

We can try to put ourselves on a more solid mathematical footing by considering actual English letter frequencies.  Here’s a list of each letter along with its relative frequency among all letters, compiled by looking at all the words listed in the Concise Oxford Dictionary (vowels are in red):

 A 8.50% B 2.07% C 4.53% D 3.38% E 11.16% F 1.81% G 2.47% H 3.00% I 7.54% J 0.20% K 1.10% L 5.49% M 3.01% N 6.65% O 7.16% P 3.17% Q 0.20% R 7.58% S 5.74% T 6.95% U 3.63% V 1.01% W 1.29% X 0.29% Y 1.78% Z 0.27%

This table tells us, for example, that by this measure around 11.16% of all letters are E’s, while around 3.63% are U’s.  In other words, E is about three times more frequent than U.  So maybe E’s should cost three times as much.  Or maybe not; I’m not trying to argue for one pricing scheme over another.

One problem with using this data to assess the cost of vowels is that it treats all words in the dictionary equally.  But is this really the best way to measure the relative frequency of letters in Wheel of Fortune?  After all, not every word in the dictionary is equally likely to appear in one of the show’s puzzles.  Moreover, the puzzles have to be simple enough that people at home can play along with the contestants, but not so simple that the contestants can solve the puzzles immediately.  These facts likely bias the frequencies of letters that appear in Wheel of Fortune puzzles.

What we really need is a database of all the puzzles that have appeared on the show that we can mine for useful information.  While complete records of the game are hard to come by, for many years one person has taken on the thankless task of recording the puzzles in the final round.

The final round works a little bit differently.  Only the player who won the most during regular gameplay competes to solve the final puzzle.  The player is given six letters for free (R, S, T, L, N, and E), and is allowed to request an additional three consonants and one vowel.  If any of these ten letters appear in the puzzle they are revealed, and the contestant then has 10 seconds to solve.

Of course, this introduces biases of its own (producers will never choose the word “restless” as the final puzzle, for example).  Nevertheless, looking at letter frequencies among nearly a decade of final round puzzles is an interesting exercise, and with the help of your friendly neighborhood spreadsheet program, is not terribly difficult.  Let’s take another look at the table above, with the dictionary relative frequencies in the middle and the Wheel of Fortune final puzzle relative frequencies on the right.

 A 8.50% 8.49% B 2.07% 3.80% C 4.53% 3.26% D 3.38% 3.43% E 11.16% 7.43% F 1.81% 3.28% G 2.47% 4.50% H 3.00% 4.69% I 7.54% 8.13% J 0.20% 0.70% K 1.10% 2.71% L 5.49% 3.57% M 3.01% 2.23% N 6.65% 4.55% O 7.16% 9.40% P 3.17% 3.34% Q 0.20% 0.37% R 7.58% 5.02% S 5.74% 3.88% T 6.95% 4.80% U 3.63% 4.40% V 1.01% 1.57% W 1.29% 2.51% X 0.29% 0.39% Y 1.78% 3.10% Z 0.27% 0.42%

If we compare how frequently each vowel appears, our original frequency data gives us the ordering E, A, I, O, U (from most to least common), while the frequency from the Wheel of Fortune final round gives the ordering O, A, I, E, U.  Does this mean O should be the most expensive vowel?

Again, not necessarily.  As expected, the relative frequencies obtained from the final round are strongly biased against the letters R, S, T, L, N, and E.  In fact, these letters account for six of the nine letters whose relative frequency decreases in moving from the first column to the second (the other three are C, M, and A).  So maybe this data isn’t particularly useful for helping us to price vowels.

It does have one fun application, though: we can use it to determine the best letters to guess if we make it to the final round.  Contestants of course want to pick letters that have a high likelihood of appearing in the puzzle.  But using the relative frequencies from the list of dictionary words can be deceiving; based on these, the three most common consonants after R, S, T, L, and N are C, D, and P, and the most common vowel after E is A.  This suggests contestants should choose C, D, P, and A for their letters.  However, based on the frequencies taken from old Wheel of Fortune episodes (arguably a better measure of what’s likely behind the board), contestants should choose  H, G, B, and O – a completely different list!

But wait, there’s more.  Since all letters in the set R, S, T, L, N, and E are revealed before the contestant has to decide what letters to pick, she has even more information to help her make a decision.  For example, suppose one E is revealed; not only does this tell you that there is only one E, it also tells you that the letters R, S, T, L, and N don’t appear at all!  This changes the relative frequencies of all the letters: for example, the relative frequency of a D nearly doubles, from 3.43% to 6.25%.  It turns out that the best choice of letters if one E appears is AB, C, D – how easy to remember!

There are any number of other ways you can slice and dice the data, but I think I’ll leave it at that for now.  If you’re more of a purist when it comes to word puzzles, then I highly recommend this article on Hangman strategy, which not only talks about how frequencies change when you know certain letters in the puzzle, but also considers factors like word length.

While this analysis may help you win big on Wheel of Fortune, it’s not great at helping you decide whether or not the way vowels are priced makes sense.  To do that, you’d need to collect data on all the puzzles, not just the ones in the final round.  If you’d like to start watching and recording hundreds of episodes of the show for this purpose, I’d encourage you to do so.  For my part, though, I think I’ll need a brief respite from the spinning wheel and flashing lights.

Teachers: want to discuss some of these ideas with your students?  Then check out our new lesson, I’d Like to Buy a Vowel!

### One Response to “I’d Like to Buy a Vowel”

1. Nick Berry says: