Wikipedia:Reference desk/Archives/Mathematics/2007 March 28

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March 28

Probability of choosing a 7-letter word from 7 random Scrabble letters

An approximation will do nicely if an accurate answer to this question isn't possible. If I use the 98 letter tiles from the English-language version of a Scrabble game (omitting the 2 blank tiles), what is the probability of selecting 7 random letters that make a 7-letter word found in an English language dictionary?

For this exercise, it can include proper names, but not words with hyphens, apostrophes etc.

The letter distribution is as follows:

• 12 tiles - E
• 9 tiles - A, I
• 8 tiles - O
• 6 tiles - N, R, T
• 4 tiles - D, L, S, U
• 3 tiles - G
• 2 tiles - B, C, F, H, M, P, V, W, Y
• 1 tile - J, K, Q, X, Z.

JackofOz 03:26, 28 March 2007 (UTC)

You can probably find what you're looking for if you google SOWPODS with some other relevant search terms. I found a list of the "high probability" seven letter words: High Probability SOWPODS Sevens - Rainwarrior 04:08, 28 March 2007 (UTC)

Actually, after checing SOWPODS in this very encyclopedia, it tells me that there are 30229 seven letter words. The probability of each of them is going to be unique, basically, which means that there's be a whole lot of calculation to do to answer your question properly. - Rainwarrior 04:11, 28 March 2007 (UTC)

Hmm. I see the problem. I had forgotten that not all 7-letter words are equally likely to be selected (and without the blank tiles, some are impossible). As a very rough approximation, how valid is dividing 30229 by 216553 (the total number of words in SOWPODS) to come up with 14%? JackofOz 04:57, 28 March 2007 (UTC)

Since not are equally likely (in fact 419 of the 31229 are impossible), that is a poor approximation. And you shouldn't divide by the number of valid words, but by the number of possible draws of tiles, and that's a bit more complicated to compute. By the way, i think i'll have an answer sometime today. :D -- Jokes Free4Me 07:59, 28 March 2007 (UTC)

Out of the remaining 30810 valid words, 5971 are repetitions, leaving only 24839 "good" draws from the 98c7 = 13,834,413,152 possible draws. However, each of the good draws have better chances of coming up than the usual "1 in N", which yields the answer of 1306623930 / 13834413152 = 9.44%. Have fun. ;) -- Jokes Free4Me 08:15, 28 March 2007 (UTC)

Trivia: The highest-chance "good" draw (although still at 0.01214%, or 1 in 8237) is for "aeinort", with the only possible SOWPODS word of OTARINE. Oh, by the way, all the numbers are for a full "deck" of tiles, at the start of the game. Later on, the numbers change, obviously depending on what has already been drawn... -- Jokes Free4Me 08:33, 28 March 2007 (UTC)

Yes, thanks, I was aware of that. I'm not talking about playing a game of Scrabble; I'm using the 98 Scrabble tiles for a different purpose. JackofOz 06:09, 30 March 2007 (UTC)

Which English language dictionary? 202.168.50.40 04:08, 28 March 2007 (UTC)

SOWPODS will do for this purpose. JackofOz 04:57, 28 March 2007 (UTC)

If dividing 30229 by 216553 gives the right result, it must be accidental. First off, this computation takes no account of the distribution of tiles. Also, suppose you ask the same question for 2-letter words. The fraction of 2-letter sowpods is 121/216553, or about 0.4%. The chances of hitting on one of these 121 words by drawing two tiles is however larger than 121/26² or 17%, which is what you'd get if the tile frequencies do not take the letter frequencies of English into account, but are all the same.

I ran a little experiment, using a list of 5579 7-letter words from an English word list after removing words that looked like inflected forms, such as singing, opposes, and lurched. Then I had an automated process repeatedly draw 7 tiles at random from the 98 tiles and check if they could make a word from the list. On 20000000 trials this gave 488115 successes. Compensating for the fact that the list contained only 5579 words, instead of all 30229 7-letter sowpods, this gives an estimated success rate of about 13.2%, since 30229/5579 × 488115/20000000 = 0.13224. The assumption is that the statistical properties of my 5579 words are quite similar to those of the 30229 7-letter sowpods. If the latter contain many "unlikely" words like ZYXXYZX, 13.2% is an overestimate. Also, the process of removing words suspected to be inflected was based on somewhat coarse criteria.  --Lambiam^Talk 08:13, 28 March 2007 (UTC)

Actually, there are quite a few "unlikely" sowpods, like ABUBBLE, ACYCLIC, BOWWOWS, TOKAMAK, ZIGZAGS, ZOOZOOS and ZYZZYVA. ;) -- Jokes Free4Me 08:33, 28 March 2007 (UTC)

None of which can be spelled using the letters from a single Scrabble set. − Twas Now ( talk • contribs • e-mail ) 15:31, 29 March 2007 (UTC)

And 31229/5579 × 488115/20000000 = 0.13661, or about 13.7%.  --Lambiam^Talk 08:18, 28 March 2007 (UTC)

Using the word list downloadable from www.scrabulous.com, I get the same result (1306623930/13834413152 = 9.44%) as Jokes reported above.  --Lambiam^Talk 11:01, 28 March 2007 (UTC)

• Thanks for this fantastic exhibition of brain power, folks. So, is the answer to my question c. 13.7% or c. 9.4%? There seems to be some disagreement above.
• I guess in a sense it's academic, since nobody is going to know all the 7-letter words in sowpods. I might pull out 7 letters that make a sowpods word, but not know that such a word exists. Or, even if I know the word, I might not recognise it from the letters in my rack. In reality, if I went first in an actual Scrabble game, the number of words at my disposal for the opening word is much less than 31,229. Some rainy day, I might go through the sowpods list and see how many I actually know, or am reasonably likely to remember. Of course, in a real game, there'd be 100 tiles, including the 2 blanks, which significantly changes the odds. I'm starting to see why such an apparently simple question does not have a simple answer. JackofOz 06:09, 30 March 2007 (UTC)

If you use the 31229 7-letter words from the sowpods list, and you draw 7 tiles from the 98 letter tiles, each of which has exactly equal probability of being drawn, then the answer is: roughly 9.444736944343%. The value 13.7% was based on an experiment using a more normal word list and then adjusting for its smaller size. This was an overestimate because, apparently, quite a few words on the sowpods list are "unlikely" or downright impossible.  --Lambiam^Talk 07:29, 30 March 2007 (UTC)

Actually, some of the top Scrabble players (as seen in Word Freak) *do* know every word in SOWPODS!--TotoBaggins 14:22, 31 March 2007 (UTC)

Lets start with the brute force method, you will want a very fast computer amd alot of time for this:

First, Get a list of the 7 letter words that are valid. (TWL06 in my case) Sort the letters in each word- ie courage would be acegoru. Lets call that an "anagram". Also count the number of each letter in the word. If the word is impossible "POPPERS" (too many Ps), toss the word out.

Second, Generate each of all the possible racks that can occur. There should be 100*99*98*97*96*95*94 or 80,678,106,432,000 possible racks. Think of it as nearly 100 to the 7th. As each rack is generated, get the anagram of the rack and discard it if none of the word anagrams matches it. If a word anagram matches it, add one to your counter of "bingo" racks.

At the end, you will have a count of the number of bingo racks that you can divide by the number of possible racks. That is the # of successes and the number of trials. Divide Success by trials for probability.

If you are not into waiting around that long, you could apply some optimization and only generate unique racks plus a number of how many ways that rack can be generated - ie if the rack is ADLSJKQ there would be 9 ways to choose an A, 4 ways to choose a D, 4 ways to choose an L... or 9*4*4*4*1*1*1 or 576 ways this same rack could occur. This will certainly speed up the process because you are comparing your words to a smaller set of racks.

More complicated than the above example is racks with repeated letters, like 3 Es for instance. Then you have to put in the number of ways you can choose 3 of the 12 Es, That would be 12*11*10 or 1320. So, if the rack was ADEEELS, it would be 9*4*(12*11*10)*4*4 or 760320 ways.

So now you get to the point where you say, Eureeka!, I can see that all I really have to do is run each valid word through and see how many racks can be generated that make that word! Way better, because then I dont have to mess around with generating all those racks. So, what you do is make an anagram of each possible word and then remove the duplicate anagrams. For example "RESHOOT", "SHOOTER", and "HOOTERS" all have the same anagam, "EHOORST". Only need one of those. Then calculate the number of possible ways to rack EHOORST - 12*2*(8*7)*6*4*6 = 193536. (Probability of getting a bingo so you can play HOOTERS is 193536/80,678,106,432,000 or 0.000000002398866). Anyway, sum the number of racks for each unique letter sorted word and you have the number of successes. The number of trials is 80,678,106,432,000. Successes over trials is the probability of a bingo.

MagRobot

Thanks, MagRobot. -- JackofOz 06:07, 7 September 2007 (UTC)

Polynomial Division

I am stuck with the following problem involving polynomials. (this is not a HW question but a teaser by my sister's math teacher that's been bugging me for a while: How can someone proceed with such a problem.

The reaminder of p(x) when divided by x+1 is 2. The reaminder of P(X) when divided by x-2 is 4. What is the reaminder p(x) by (x+1).(x-2) (the product)?

When i first read the problem i directly thought of modular arithematics,but i don't think the solution involves modular arithematics because my sister is still in high school and she is not familiar with it.

Hisham1987 15:38, 28 March 2007 (UTC)

Well, it's related to modular arithmetic (sort of). You are being asked for a special case of the (generalised) Chinese remainder theorem. However, this case is a simple one so no high-powered machinery is needed. We have

p=(x+1)f + 2

p=(x-2)g + 4

for some polys f,g. We want

p=(x+1)(x-2)h + r

try multiplying the two equations we've got by appropriate factors and fiddling. Algebraist 16:19, 28 March 2007 (UTC)

Alternatively, you could write the remainder r=ax+b and use r(-1)=2 and r(2)=4 to solve for a and b. 194.222.88.44 16:32, 28 March 2007 (UTC)

If this is a teaser, we're meant to have fun with it.

The question applies to a family of polynomials; can we find one? If we know one beautiful fact, this is easy. That fact is, the remainder of a polynomial P(x) when divided by a linear polynomial, x−c, is just P(c). (This is not hard to prove, if you'd like to try.) So we need only fit a "curve" through two data points, for which a line, P(x) = ax+b, will suffice. Thus we solve the following two equations for a and b.

${\begin{aligned}a(-1)+b&{}=2\\a(2)+b&{}=4\end{aligned}}$

Anything that is true in general must be true in a particular case. We can find a P that satisfies the remainder conditions; what does this case tell us?

This was the first part of our fun. The second part is to connect what we learned with this experiment to what others have mentioned. We would like to do this both for curiosity, and because the trick we used will not work if we divide by polynomials of higher degree. --KSmrq^T 18:04, 28 March 2007 (UTC)

Here's a hint... denote your polynomial as $p(x)=f(x)(x+1)(x-2)+ax+b$ and see what happens. Cheers --Hirak 99 21:40, 28 March 2007 (UTC)

Probability of a "chain" of outcomes of certain length, in a long chain of events

I am not familiar with all the proper wordings and notation, so hopefully the subject isn't too confusing.

What I would like to learn is of a method to calculate the following kind of problem:

I have a series of 1,000 "events" For each such event, the outcome will be HEAD with 24.5% certainty, and TAIL with 75.5% certainty. The outcome for each event is independent of any "history".

1. What is the likelihood that

a) The longest "chain" of consecutive HEADs will be exactly 12?

b) The longest "chain" of consecutive HEADs will be 10 or longer?

2) What is the most likely longest chain of HEADs?

3) Given the above certainities, how many events have to occur for the probability of having a chain of HEADs 7 or longer being .75 or greater?

NB: I do not wish for the answers to these particular questions, as they are just examples I just made up, but rather learn of the reasoning behind solving such problems. If the methods used are fairly simple, such as a few formulas where I'd put in the different values, I would still greatly appreciate any links to articles on the underlying math. 81.20.147.194 17:37, 28 March 2007 (UTC)

I believe the field of statistics addressing such questions is called scan statistics, which at present is a redlink. As far as I know (which isn't a lot), such problems are addressed through Monte Carlo simulation. The complexity arises from the fact that, even though the individual event is independent of the history, the composite events that you are asking about are not, and an exact calculation would require examining each of the 2¹⁰⁰⁰ possibible outcomes, a computationally prohibitive task. I'd love some input on this topic from someone better informed than I am. --NorwegianBlue ^talk 19:43, 28 March 2007 (UTC)

Look up Bernoulli distribution, Bernoulli process, geometric distribution, negative binomial distribution (and links therein). (Igny 21:42, 28 March 2007 (UTC))

Yes those distributions are useful, do look them up. However, unfortunately they are only good for infinite case, and hence will help you get an approximate solution. See http://mathworld.wolfram.com/Run.html for more detailed information on finite case. Cheers --Hirak 99 22:09, 28 March 2007 (UTC)

?????????

5+6+5+4+7+9+54+3+54+56+67+76+56+56+67+76+67+67+76+67+54+54+54+54+54+45*3 —Preceding unsigned comment added by 86.129.225.138 (talk • contribs)

Paste than into Google, and it'll give you an answer. This reference desk isn't a calculator ;) -- Consumed Crustacean (talk) 20:30, 28 March 2007 (UTC)

1283. --Ķĩřβȳ♥♥♥ŤįɱéØ 08:55, 29 March 2007 (UTC)

ith root of i

What is the ith root of i? —Preceding unsigned comment added by 202.168.50.40 (talk • contribs)

As described in Imaginary unit#i and Euler's formula, consider that $ln(i)={\frac {i\pi }{2}}$. Then it follows that $i^{\frac {1}{i}}=e^{\frac {ln(i)}{i}}=e^{\frac {\pi }{2}}$. Dugwiki 23:02, 28 March 2007 (UTC)

$e^{\frac {\pi }{2}}$ is certainly one of the ith roots of i. So are $e^{\frac {5\pi }{2}}$ or $e^{\frac {-3\pi }{2}}$, or $e^{\frac {(4n+1)\pi }{2}}$ for any integer n. In other words, the equation $x^{i}=i$ has an infinite number of solutions. Gandalf61 10:16, 29 March 2007 (UTC)

Thanks, Gandalf, I stand somewhat corrected. The periodicity of raising to the i'th power follows again from Euler's formula $e^{ix}=\cos \;x+i\sin \;x\,$. Dugwiki 17:38, 29 March 2007 (UTC)

Alas, even more care is required, since $x^{i}$ is also multi-valued. So Gandalf's second observation is better stated "there are infinitely many x such that one of the values of $x^{i}$ is i". -- Meni Rosenfeld (talk) 18:56, 29 March 2007 (UTC)

Complicated stuff, which I suppose is why it's called "Complex analysis" Dugwiki 19:12, 29 March 2007 (UTC)

I'm curious (but I guess you already knew that). What is the numerical value of $e^{\frac {\pi }{2}}$ to 4 decimal places? JackofOz 06:14, 30 March 2007 (UTC)

4.81047738097. —Bkell (talk) 07:28, 30 March 2007 (UTC)

Thanks. JackofOz 08:45, 30 March 2007 (UTC)

Jack, now I'm curious. Do you not have a calculator in your immediate vicinity (if not a physical one, which is always good, at least "Windows Calculator" or the like)? -- Meni Rosenfeld (talk) 15:01, 30 March 2007 (UTC)

Windows Calculator gives 4.8104773809653516554730356667038 --Ķĩřβȳ♥♥♥ŤįɱéØ 20:45, 30 March 2007 (UTC)

I have a Windows calculator, but I don't know how to work out a non-algebraic power of a number. JackofOz 09:54, 31 March 2007 (UTC)

Has it the exp() function? —Tamfang 20:58, 2 April 2007 (UTC)

Yes (sorry for apparently ignoring the question - it got buried in my in tray). -- JackofOz 06:09, 7 September 2007 (UTC)

Wow, talk about reviving an old archive! Anyway, the "Exp" button I see in my Windows calculator is something else (used for scientific notation), not what we want. The exponential function $e^{x}$ can be obtained by checking "Inv" and clicking "ln". So the way to calculate this is "pi / 2 = Inv ln =". Another variation is "2.718281828 x^y ( 3.141592653 / 2 ) =". -- Meni Rosenfeld (talk) 11:17, 7 September 2007 (UTC)

Eeeeexcellent. Thank you, Meni. -- JackofOz 14:34, 8 September 2007 (UTC)