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− | The NASPA Zyzzyva FAQ, as with everything Zyzzyva-related, was originally written by Michael Thelen and then transferred to NASPA in 2014. | + | The '''NASPA Zyzzyva FAQ''', as with everything [[NASPA Zyzzyva|Zyzzyva]]-related, was originally written by [[Michael Thelen]] and then transferred to NASPA in 2014. |
− | + | Please note that this FAQ describes the desktop version of NASPA Zyzzyva. For information about the new NASPA Zyzzyva Mobile app, please visit [[NASPA Zyzzyva Mobile|this page]]. | |
− | == | + | == General == |
− | + | === Why is the program called Zyzzyva? === | |
− | == Why is Zyzzyva available for free? == | + | Michael Thelen writes: I wanted to pick a fun word that was somewhat unique. I knew zyzzyva was one of the most improbable words to be played in a Scrabble game, and I liked the sound of it. Later on, after I had picked the name, I realized that since zyzzyva was also the last word in alphabetical order (it has since been replaced by zzz), I could call Zyzzyva the last word in word study. I wish I could say I was smart enough to come up with that slogan from the start. |
+ | |||
+ | === Why is NASPA involved with Zyzzyva? === | ||
+ | |||
+ | With the release of the OTCWL2014 and OSPD5 lexicons in 2014, Hasbro, Inc. asked Michael Thelen to sign a license agreement that would bind him to prevent the leakage of the lists included with the Zyzzyva distribution. As a result, he decided to sell the product to NASPA, and the new NASPA Zyzzyva Committee began work to change the product so that it would comply with the terms of license, and to tie its availability to NASPA membership. | ||
+ | |||
+ | Beginning with version 3.1.0, NASPA Zyzzyva includes Collins lexicons under license, allowing "bilexical" players to study the current international and North American tournament lexicons and the differences between them. | ||
+ | |||
+ | Version 3.2.0 removed protection from the NASPA tournament/club and school lexicons, and version 3.2.1 removed protection from the Collins lexicons. This meant that a license was no longer required for NASPA members to use these lexicons, except to perform a one-time decryption of any existing quiz statistics databases. | ||
+ | |||
+ | Version 3.3.0 removed all vestiges of these licensing and decryption features. As of August 2021, NASPA Games and its activities are neither endorsed by nor affiliated with Hasbro or Mattel. | ||
+ | |||
+ | === Why is Zyzzyva available for free? === | ||
Michael Thelen writes: I created Zyzzyva because I wanted a program for studying SCRABBLE words. There are several programs available, but all of them either lack features I wanted or do not run on Linux. I therefore took it upon myself to create the program I wanted, and Zyzzyva is the result. | Michael Thelen writes: I created Zyzzyva because I wanted a program for studying SCRABBLE words. There are several programs available, but all of them either lack features I wanted or do not run on Linux. I therefore took it upon myself to create the program I wanted, and Zyzzyva is the result. | ||
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NASPA adds: We appreciate Michael’s efforts in developing Zyzzyva and are committed to continuing to make it freely available to the SCRABBLE community. For licensing reasons, however, and the need to support proprietary word lists, Zyzzyva is now closed-source. | NASPA adds: We appreciate Michael’s efforts in developing Zyzzyva and are committed to continuing to make it freely available to the SCRABBLE community. For licensing reasons, however, and the need to support proprietary word lists, Zyzzyva is now closed-source. | ||
− | == | + | === I love Zyzzyva so much that I would like to send you giant piles of money. How can I do that? === |
+ | |||
+ | Michael Thelen originally wrote: That is so nice of you! I'm always very happy to receive donations of any amount. | ||
+ | |||
+ | NASPA writes: You can support Zyzzyva development and our other activities by [http://www.scrabbleplayers.org/cgi-bin/join.pl joining] NASPA or [https://www.scrabbleplayers.org/cgi-bin/renew.pl keeping] your membership current. | ||
+ | |||
+ | If you are feeling philanthropic, why not consider a donation to our [https://www.naspafoundationforyouthliteracy.org/ Foundation for Youth Literacy]? | ||
+ | |||
+ | === When will NASPA Zyzzyva be available for my platform? === | ||
+ | |||
+ | NASPA Zyzzyva is currently available for the Windows, macOS, and Linux platforms. | ||
+ | |||
+ | As of version 3.3.0, there is a version for mobile devices, too! When NASPA took over Zyzzyva development, the iOS version was already two iOS updates obsolete. We have reinvented it as an app for iOS and Android, with much the same user interface as that pre-2014 app, but built upon the same base source code as the desktop version so that it can be developed in parallel and new features added. For more information on NASPA Zyzzyva Mobile, click [[NASPA Zyzzyva Mobile|here]]. | ||
+ | |||
+ | == Supported Word Lists == | ||
+ | |||
+ | === How do I use word lists added in the latest version of NASPA Zyzzyva?=== | ||
+ | |||
+ | When you install a new version of NASPA Zyzzyva, it does not change the list of lexicons to include any newly added word lists. | ||
+ | |||
+ | Follow these instructions to make a word list available in NASPA Zyzzyva: | ||
+ | |||
+ | * Choose Edit > Preferences > Load Lexicons > Edit... and check the lexicons that you want to use. | ||
+ | * Rebuild databases when prompted. This step may take several minutes if you have many lexicons installed. When it completes, you’re done! | ||
+ | ** If not prompted, you can choose Tools > Rebuild Database... to start the process. | ||
+ | |||
+ | === Where can I find word lists to use with NASPA Zyzzyva? === | ||
+ | |||
+ | If the word list is not [[Zyzzyva#Current_Version | included]] in NASPA Zyzzyva, you may be able to load a text file version of the word list as a custom lexicon. | ||
+ | |||
+ | It depends on the word list, but you should in general ask the copyright holder for permission to use an electronic copy of the word list. | ||
+ | |||
+ | If you need help formatting it for use with Zyzzyva, please feel free to contact [mailto:zyzzyva@scrabbleplayers.org us]. | ||
− | + | == Using Zyzzyva == | |
− | == How | + | === How do I print a list of words? === |
− | You can | + | There is no print functionality built into Zyzzyva. You can, however, right-click (Control-click on a Mac) on any word list (e.g., Search results) and save it in plain text format. Then manipulate the list and print it from a text editor program, like Notepad on Windows. |
+ | |||
+ | === Why do I not see definitions in search results for a NASPA Word List that I just loaded? === | ||
+ | |||
+ | After you load a word list, you must build the database for that word list. Go to Tools -> Rebuild Database and choose the new word list or all word lists. | ||
+ | |||
+ | Also, make sure that the Show Definitions option is checked in Word Table Preferences under the Edit menu. | ||
+ | |||
+ | === When viewing search results, what are the marks at the beginning or end of certain words? === | ||
+ | |||
+ | Those marks denote inner hooks - they mean that either the first or last letter of the word can be dropped to form another valid word, depending on where the mark appears. If the mark appears at the beginning of the word, then the first letter can be dropped. If the mark appears at the end of the word, then the last letter can be dropped. | ||
+ | |||
+ | You can choose whether to display a hyphen or a bullet character to denote inner hooks in Word Table Preferences under the Edit menu. | ||
+ | |||
+ | === Why did Zyzzyva mark all my answers as missed when I actually got them all correct? === | ||
− | The | + | Zyzzyva only marks all responses as missed if you actually entered all the correct responses, but also provided an incorrect response. The reason for marking all responses as missed is because (currently) Zyzzyva has no other way of marking the question as missed without marking at least one response as missed. So instead of picking a random response to mark as missed, it marks them all. |
− | + | This only happens if you have the 'Mark question as missed after an incorrect response' setting activated, but not 'End question after an incorrect response'. Technically, it could also happen if you have the 'End question after an incorrect response' setting activated, but not 'End question after all correct responses', and the incorrect response is the last one you give. | |
− | + | == Cardboxes and Quizzes == | |
− | + | === How does the Cardbox system work? === | |
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− | + | The basic idea is that you must first tell Zyzzyva which words you'd like it to keep track of. You generally do this by performing a search, then right-clicking the search results, and choosing Add to Cardbox. Then Zyzzyva keeps track of those words and automatically quizzes you on the ones it thinks you need to see each day. When you miss a word frequently, Zyzzyva will show it to you more often. | |
− | + | Check out the [[Cardboxing 101]] and [[Cardboxing 201]] articles to get an idea of how you can use Cardboxing for word study. | |
− | + | Also, the Help documentation provides more detailed instructions on the use of this feature. | |
− | + | === How do I retain my quiz statistics and cardbox data when I want to study a different word list? === | |
− | + | Cardboxes are separate for each word list. You can, however, import quiz statistics and cardbox data from one word list to another. | |
− | + | * Choose File > Cardbox. | |
+ | * Select the Lexicon - for example, NWL2023 - and Quiz Type - usually Anagrams. | ||
+ | * Choose Import... | ||
+ | * Select the lexicon from which you want to import data - for example, NWL2020 - and click OK. | ||
− | == How do I remove all my cardbox data? == | + | === How do I remove all my cardbox data? === |
Do the following: | Do the following: | ||
* Open a Search tab. | * Open a Search tab. | ||
− | * Do a search that will find all possible words in your cardbox. Probably something like | + | * Do a search that will find all possible words in your cardbox. Probably something like: Length: Min 2, Max 9. |
− | * Right-click the search results and choose | + | * Right-click the search results and choose Remove list from Cardbox. |
+ | <br/> | ||
+ | All your cardbox data will be removed from the system, but your historical quiz data will be retained. | ||
+ | |||
+ | == Portability of Zyzzyva Data == | ||
+ | |||
+ | === How can I transfer my Zyzzyva data from one computer to another? === | ||
+ | |||
+ | You can transfer all your data to a new computer by copying your data directory. You can find out where your Zyzzyva data directory is by opening the Preferences window in Zyzzyva and looking at the Data Directory setting. Make a copy of this directory, and move it to wherever you like, on the new computer. Then, after you install Zyzzyva on the new computer, open the Preferences window and change the Data Directory setting to point to wherever you put your data directory from the original computer. | ||
− | + | The transfer should work regardless of the operating systems used on the two computers. For example, you should be able to transfer your data directory from a Windows computer to a Mac, or vice versa. | |
+ | |||
+ | If you have access to the Internet, you can tell Zyzzyva to synchronize its data directory with cloud storage, allowing you to use multiple computers and mobile devices to study using the same data. See the Help documentation for more details about data synchronization. | ||
+ | |||
+ | === How can I change the location of my data directory? === | ||
+ | |||
+ | Always be careful when doing this, so you don't lose important data that you've been working on. | ||
+ | |||
+ | While Zyzzyva is running, open the Preferences window and change the Data Directory setting to the new location. (Newer versions of Zyzzyva offer a Data Directory Wizard to help you with this.) Use the option to Move data after saving preferences. When you save preferences, Zyzzyva will close all open tabs, move the data directory, and restart itself in the new location. | ||
+ | |||
+ | === Can I keep my Zyzzyva data on a USB thumb drive? === | ||
− | + | Absolutely. This may be a good option for anyone wanting to use Zyzzyva on multiple computers without having to copy data directories back and forth or sign up for cloud storage. Simply copy your data directory onto the USB drive (using the process described to transfer Zyzzyva data from one computer to another). Then open the Preferences window and change the Data Directory setting to point to the data directory on the USB drive. | |
− | + | == Word Probability == | |
− | == How does Zyzzyva calculate the probability of a word? == | + | === How does Zyzzyva calculate the probability of a word? === |
− | + | There isn't a handy formula, but it's basically a straight calculation of the number of possible combinations to get a particular set of letters from a full bag, including blanks. Separate combination values are calculated for drawing zero, one, or two blanks, and then the numbers of combinations are added together to get the total number of ways of drawing a word. | |
− | == Are you sure the probability calculation is accurate? It seems unintuitive to me. == | + | === Are you sure the probability calculation is accurate? It seems unintuitive to me. === |
Yep, it really is correct. The calculation of probability is notorious for violating human intuition. One useful guideline to remember is that doubled or tripled letters make a particular letter combination much more improbable than you might think. That's because the number of ways to draw two identical letters out of the bag is generally much lower than the number of ways to draw two different letters out of the bag. | Yep, it really is correct. The calculation of probability is notorious for violating human intuition. One useful guideline to remember is that doubled or tripled letters make a particular letter combination much more improbable than you might think. That's because the number of ways to draw two identical letters out of the bag is generally much lower than the number of ways to draw two different letters out of the bag. | ||
− | == Are you really sure about the probabilities? I need to see more examples. == | + | === Are you really sure about the probabilities? I need to see more examples. === |
− | Consider BAILERS versus BELIERS. BAILERS is a fair amount more probable than BELIERS because of the doubled Es. However, it seems intuitive to say, "If BEILRS has been drawn from a full bag, then there are 9 As and 11 Es in the remaining pool. The E is the more likely draw, which means BELIERS should be more probable than BAILERS." | + | Consider BAILERS versus BELIERS. BAILERS is a fair amount more probable than BELIERS because of the doubled Es in BELIERS. However, it seems intuitive to say, "If BEILRS has been drawn from a full bag, then there are 9 As and 11 Es in the remaining pool. The E is the more likely draw, which means BELIERS should be more probable than BAILERS." |
While it is true that drawing a single E is more likely than drawing a single A, the math for calculating the probability of a word is more complicated than that. You can't just start calculating with the assumption that BEILRS has been drawn; those letters themselves factor into the calculation, and they are not independent events. In this case, we can reduce the difference between BAILERS and BELIERS to the difference between drawing AE or EE when drawing two tiles from the bag. | While it is true that drawing a single E is more likely than drawing a single A, the math for calculating the probability of a word is more complicated than that. You can't just start calculating with the assumption that BEILRS has been drawn; those letters themselves factor into the calculation, and they are not independent events. In this case, we can reduce the difference between BAILERS and BELIERS to the difference between drawing AE or EE when drawing two tiles from the bag. | ||
− | If you draw two tiles out of a full bag, the probability of drawing two Es is much lower than the probability of drawing an A and an E. That is because there are 9 x 12 = 108 ways to draw an A and an E if you draw two tiles. But there are only (12 x 11) / 2 = 66 ways to draw two Es if you draw two tiles. This number of combinations is called "12 choose 2", and is explained in more detail in this Wikipedia article about combinations. | + | If you draw two tiles out of a full bag, the probability of drawing two Es is much lower than the probability of drawing an A and an E. That is because there are 9 x 12 = 108 ways to draw an A and an E if you draw two tiles. But there are only (12 x 11) / 2 = 66 ways to draw two Es if you draw two tiles. This number of combinations is called "12 choose 2", and is explained in more detail in this [https://en.wikipedia.org/wiki/Combination Wikipedia article] about combinations. |
To reduce this to a simple example where it's easy to enumerate all the possibilities, consider a 4-tile bag containing only AAEE. I'll label the tiles A1, A2, E1, E2 for convenience. There are 6 possible ways to draw two unordered tiles: | To reduce this to a simple example where it's easy to enumerate all the possibilities, consider a 4-tile bag containing only AAEE. I'll label the tiles A1, A2, E1, E2 for convenience. There are 6 possible ways to draw two unordered tiles: | ||
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E1 E2 = EE | E1 E2 = EE | ||
− | As you can see, AE is drawn 4/6 times, while AA is drawn only 1/6, and EE is drawn only 1/6 as well. Even though A and E are equally likely to be drawn if you are only drawing one tile, if you're drawing two tiles it is much more likely to draw one of each than to draw duplicates. | + | As you can see, AE is drawn 4/6 times, while AA is drawn only 1/6, and EE is drawn only 1/6 as well. Even though A and E are equally likely to be drawn if you are only drawing one tile, if you're drawing two tiles, it is much more likely to draw one of each than to draw duplicates. |
− | As another example, consider a bag where E is more likely than A, as is the case in the full | + | As another example, consider a bag where E is more likely than A, as is the case in the full SCRABBLE bag. Consider a 6-tile bag containing AAEEEE, with tiles labeled A1, A2, E1, E2, E3, E4. There are 15 possible ways to draw two unordered tiles: |
A1 A2 = AA | A1 A2 = AA | ||
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E3 E4 = EE | E3 E4 = EE | ||
− | AA occurs 1/15 times, AE occurs 8/15 times, EE occurs 6/15 times. Even though E is twice as likely as A, AE is still more likely than EE! This is the same basic principle that also causes words with duplicate letters to be drawn with lower probability from a full | + | AA occurs 1/15 times, AE occurs 8/15 times, EE occurs 6/15 times. Even though E is twice as likely as A, AE is still more likely than EE! This is the same basic principle that also causes words with duplicate letters to be drawn with lower probability from a full SCRABBLE bag. |
Complicating the math even further is the fact that Zyzzyva calculates probabilities using a full 100-tile bag including two blanks. That may account for differences (usually minor) with other probabilities you may have seen calculated elsewhere, many of which use a 98-tile bag with no blanks. | Complicating the math even further is the fact that Zyzzyva calculates probabilities using a full 100-tile bag including two blanks. That may account for differences (usually minor) with other probabilities you may have seen calculated elsewhere, many of which use a 98-tile bag with no blanks. | ||
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The NASPA Zyzzyva FAQ, as with everything Zyzzyva-related, was originally written by Michael Thelen and then transferred to NASPA in 2014.
Please note that this FAQ describes the desktop version of NASPA Zyzzyva. For information about the new NASPA Zyzzyva Mobile app, please visit this page.
Michael Thelen writes: I wanted to pick a fun word that was somewhat unique. I knew zyzzyva was one of the most improbable words to be played in a Scrabble game, and I liked the sound of it. Later on, after I had picked the name, I realized that since zyzzyva was also the last word in alphabetical order (it has since been replaced by zzz), I could call Zyzzyva the last word in word study. I wish I could say I was smart enough to come up with that slogan from the start.
With the release of the OTCWL2014 and OSPD5 lexicons in 2014, Hasbro, Inc. asked Michael Thelen to sign a license agreement that would bind him to prevent the leakage of the lists included with the Zyzzyva distribution. As a result, he decided to sell the product to NASPA, and the new NASPA Zyzzyva Committee began work to change the product so that it would comply with the terms of license, and to tie its availability to NASPA membership.
Beginning with version 3.1.0, NASPA Zyzzyva includes Collins lexicons under license, allowing "bilexical" players to study the current international and North American tournament lexicons and the differences between them.
Version 3.2.0 removed protection from the NASPA tournament/club and school lexicons, and version 3.2.1 removed protection from the Collins lexicons. This meant that a license was no longer required for NASPA members to use these lexicons, except to perform a one-time decryption of any existing quiz statistics databases.
Version 3.3.0 removed all vestiges of these licensing and decryption features. As of August 2021, NASPA Games and its activities are neither endorsed by nor affiliated with Hasbro or Mattel.
Michael Thelen writes: I created Zyzzyva because I wanted a program for studying SCRABBLE words. There are several programs available, but all of them either lack features I wanted or do not run on Linux. I therefore took it upon myself to create the program I wanted, and Zyzzyva is the result.
NASPA adds: We appreciate Michael’s efforts in developing Zyzzyva and are committed to continuing to make it freely available to the SCRABBLE community. For licensing reasons, however, and the need to support proprietary word lists, Zyzzyva is now closed-source.
Michael Thelen originally wrote: That is so nice of you! I'm always very happy to receive donations of any amount.
NASPA writes: You can support Zyzzyva development and our other activities by joining NASPA or keeping your membership current.
If you are feeling philanthropic, why not consider a donation to our Foundation for Youth Literacy?
NASPA Zyzzyva is currently available for the Windows, macOS, and Linux platforms.
As of version 3.3.0, there is a version for mobile devices, too! When NASPA took over Zyzzyva development, the iOS version was already two iOS updates obsolete. We have reinvented it as an app for iOS and Android, with much the same user interface as that pre-2014 app, but built upon the same base source code as the desktop version so that it can be developed in parallel and new features added. For more information on NASPA Zyzzyva Mobile, click here.
When you install a new version of NASPA Zyzzyva, it does not change the list of lexicons to include any newly added word lists.
Follow these instructions to make a word list available in NASPA Zyzzyva:
If the word list is not included in NASPA Zyzzyva, you may be able to load a text file version of the word list as a custom lexicon.
It depends on the word list, but you should in general ask the copyright holder for permission to use an electronic copy of the word list.
If you need help formatting it for use with Zyzzyva, please feel free to contact us.
There is no print functionality built into Zyzzyva. You can, however, right-click (Control-click on a Mac) on any word list (e.g., Search results) and save it in plain text format. Then manipulate the list and print it from a text editor program, like Notepad on Windows.
After you load a word list, you must build the database for that word list. Go to Tools -> Rebuild Database and choose the new word list or all word lists.
Also, make sure that the Show Definitions option is checked in Word Table Preferences under the Edit menu.
Those marks denote inner hooks - they mean that either the first or last letter of the word can be dropped to form another valid word, depending on where the mark appears. If the mark appears at the beginning of the word, then the first letter can be dropped. If the mark appears at the end of the word, then the last letter can be dropped.
You can choose whether to display a hyphen or a bullet character to denote inner hooks in Word Table Preferences under the Edit menu.
Zyzzyva only marks all responses as missed if you actually entered all the correct responses, but also provided an incorrect response. The reason for marking all responses as missed is because (currently) Zyzzyva has no other way of marking the question as missed without marking at least one response as missed. So instead of picking a random response to mark as missed, it marks them all.
This only happens if you have the 'Mark question as missed after an incorrect response' setting activated, but not 'End question after an incorrect response'. Technically, it could also happen if you have the 'End question after an incorrect response' setting activated, but not 'End question after all correct responses', and the incorrect response is the last one you give.
The basic idea is that you must first tell Zyzzyva which words you'd like it to keep track of. You generally do this by performing a search, then right-clicking the search results, and choosing Add to Cardbox. Then Zyzzyva keeps track of those words and automatically quizzes you on the ones it thinks you need to see each day. When you miss a word frequently, Zyzzyva will show it to you more often.
Check out the Cardboxing 101 and Cardboxing 201 articles to get an idea of how you can use Cardboxing for word study.
Also, the Help documentation provides more detailed instructions on the use of this feature.
Cardboxes are separate for each word list. You can, however, import quiz statistics and cardbox data from one word list to another.
Do the following:
All your cardbox data will be removed from the system, but your historical quiz data will be retained.
You can transfer all your data to a new computer by copying your data directory. You can find out where your Zyzzyva data directory is by opening the Preferences window in Zyzzyva and looking at the Data Directory setting. Make a copy of this directory, and move it to wherever you like, on the new computer. Then, after you install Zyzzyva on the new computer, open the Preferences window and change the Data Directory setting to point to wherever you put your data directory from the original computer.
The transfer should work regardless of the operating systems used on the two computers. For example, you should be able to transfer your data directory from a Windows computer to a Mac, or vice versa.
If you have access to the Internet, you can tell Zyzzyva to synchronize its data directory with cloud storage, allowing you to use multiple computers and mobile devices to study using the same data. See the Help documentation for more details about data synchronization.
Always be careful when doing this, so you don't lose important data that you've been working on.
While Zyzzyva is running, open the Preferences window and change the Data Directory setting to the new location. (Newer versions of Zyzzyva offer a Data Directory Wizard to help you with this.) Use the option to Move data after saving preferences. When you save preferences, Zyzzyva will close all open tabs, move the data directory, and restart itself in the new location.
Absolutely. This may be a good option for anyone wanting to use Zyzzyva on multiple computers without having to copy data directories back and forth or sign up for cloud storage. Simply copy your data directory onto the USB drive (using the process described to transfer Zyzzyva data from one computer to another). Then open the Preferences window and change the Data Directory setting to point to the data directory on the USB drive.
There isn't a handy formula, but it's basically a straight calculation of the number of possible combinations to get a particular set of letters from a full bag, including blanks. Separate combination values are calculated for drawing zero, one, or two blanks, and then the numbers of combinations are added together to get the total number of ways of drawing a word.
Yep, it really is correct. The calculation of probability is notorious for violating human intuition. One useful guideline to remember is that doubled or tripled letters make a particular letter combination much more improbable than you might think. That's because the number of ways to draw two identical letters out of the bag is generally much lower than the number of ways to draw two different letters out of the bag.
Consider BAILERS versus BELIERS. BAILERS is a fair amount more probable than BELIERS because of the doubled Es in BELIERS. However, it seems intuitive to say, "If BEILRS has been drawn from a full bag, then there are 9 As and 11 Es in the remaining pool. The E is the more likely draw, which means BELIERS should be more probable than BAILERS."
While it is true that drawing a single E is more likely than drawing a single A, the math for calculating the probability of a word is more complicated than that. You can't just start calculating with the assumption that BEILRS has been drawn; those letters themselves factor into the calculation, and they are not independent events. In this case, we can reduce the difference between BAILERS and BELIERS to the difference between drawing AE or EE when drawing two tiles from the bag.
If you draw two tiles out of a full bag, the probability of drawing two Es is much lower than the probability of drawing an A and an E. That is because there are 9 x 12 = 108 ways to draw an A and an E if you draw two tiles. But there are only (12 x 11) / 2 = 66 ways to draw two Es if you draw two tiles. This number of combinations is called "12 choose 2", and is explained in more detail in this Wikipedia article about combinations.
To reduce this to a simple example where it's easy to enumerate all the possibilities, consider a 4-tile bag containing only AAEE. I'll label the tiles A1, A2, E1, E2 for convenience. There are 6 possible ways to draw two unordered tiles:
A1 A2 = AA A1 E1 = AE A1 E2 = AE A2 E1 = AE A2 E2 = AE E1 E2 = EE
As you can see, AE is drawn 4/6 times, while AA is drawn only 1/6, and EE is drawn only 1/6 as well. Even though A and E are equally likely to be drawn if you are only drawing one tile, if you're drawing two tiles, it is much more likely to draw one of each than to draw duplicates.
As another example, consider a bag where E is more likely than A, as is the case in the full SCRABBLE bag. Consider a 6-tile bag containing AAEEEE, with tiles labeled A1, A2, E1, E2, E3, E4. There are 15 possible ways to draw two unordered tiles:
A1 A2 = AA A1 E1 = AE A1 E2 = AE A1 E3 = AE A1 E4 = AE A2 E1 = AE A2 E2 = AE A2 E3 = AE A2 E4 = AE E1 E2 = EE E1 E3 = EE E1 E4 = EE E2 E3 = EE E2 E4 = EE E3 E4 = EE
AA occurs 1/15 times, AE occurs 8/15 times, EE occurs 6/15 times. Even though E is twice as likely as A, AE is still more likely than EE! This is the same basic principle that also causes words with duplicate letters to be drawn with lower probability from a full SCRABBLE bag.
Complicating the math even further is the fact that Zyzzyva calculates probabilities using a full 100-tile bag including two blanks. That may account for differences (usually minor) with other probabilities you may have seen calculated elsewhere, many of which use a 98-tile bag with no blanks.
This page was last edited on 16 October 2024, at 11:18. Privacy policy
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