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Mike Haven
09-14-2002, 09:04 PM
In the 911 thread cardshark asked: "If I you roll a die, the odds of rolling, lets say 3, are 1/6. How about if you are about to roll the die six times, and I make a bet that you will roll 3 on the third roll; are the odds still 1/6?"

yes, is the answer

but here's an interesting one where the odds of future events can be calculated and do change (and there's a bet in it somewhere!)

if someone opened a book and pointed randomly at one letter on a page and asked you to guess whether it was a vowel or a consonant obviously you would have to bet that it was a consonant, as you know, instinctively, there are more consonants than vowels in a piece of writing (probably about 57% to 43% on average)

now, if you could talk this person into telling you what the letter before the one you have to guess was, things would change dramatically

a large study (markov's) showed that a vowel is followed by a consonant 87% of the time and a consonant is followed by a vowel 66% of the time!

hence, if you are told the previous letter was an "n" for example, you would gaze into the distance for a few seconds, and then "guess" vowel, with a 66% chance of being correct - well worth trying to get an evens bet out of your mark with such a high likelihood of your winning

here's where it gets even more interesting

because you know the percentages as above, you can calculate the chances of each of the NEXT letters being vowels or consonants!

for example, if the announced letter is an "e" what is the chance of the next-but-one letter also being a vowel?

there are only two patterns that would give this result: v-v-v and v-c-v

from our percentages above v-v-v will occur 13% x 13% = 1.7% of the time and v-c-v will occur 87% x 66% = 57.4% of the time

so the next-but-one letter to an "e" will be another vowel 1.7% + 57.4% = 59.1% of the time!

by the same type of calculations you will find that the letter three steps ahead from a vowel will be a consonant 65% of the time or three steps ahead from a consonant the letter will be a vowel 50% of the time

this could be the best bet to offer to disguise your winning strategy

every time he calls a vowel bet that the letter three steps ahead is a consonant and every time he calls a consonant call the opposite of your previous answer to make it sound like you are betting randomly (you ARE betting randomly at this point as it is an evens bet - you are making your money on the 65% consonant bet)

cardshark
09-15-2002, 11:12 AM
Mike, very interesting post. With your permission, I would lie to use your post on my BB. I would of course give credit to you and link it to this 2+2 forum but only if you say its OK. If however you feel its not appropriate, I will understand. My BB deals with cheating and scams so this is why I find your post fit.

Anyway, I gave my die-roll question some thought after I asked the question and before I had the chance to read your answer. I also came up with the conclusion that the odds are 1/6 even if I attempt to predict the outcome of the roll tied to another future event.

However, this is how I came to this conclusion. Id like to share my line of thought and you tell me if Im right.

We were talking about rolling one die 6 times and predicting that it will roll 3 on the 3rd roll. To better help me with my logic I assumed that this prediction should be the same as having 6 dice each of a different color (lets say) - and I place a bet that the white one will roll number 3. In this case it is very clear to me that the other dice have just as much relevance to my bet as the millions of other dice that may be rolling at the same time in other parts of the world. Meaning - no relevance whatsoever. I am placing my bet on the white die and my odds are 1/6.

So, am I correct in thinking that the 6-dice example is the same as the 6-roll example?

Peter
09-15-2002, 04:29 PM
Yes Igor, the six-dice example is the same as the six-rolls example for the following reason: with the six dice, you found it easy to understand that the other dice had no influence at all on the die you were betting on. Since dice have no memory, previous rolls also have no influence on the next roll of one single die, just as the previous spin of a roulette wheel has no influence on the next spin.
Therefore, the probability of getting a specific number in the third out of six rolls is the same as the probability of rolling that number with a die that is rolled together with five different colored dice.

Peter

Peter
09-15-2002, 04:39 PM
Interesting post. In the study you mention, did they only look at the american language or also at other languages?

I think the subject of the book that is used would also influence the outcome. To give an oversimplified example, if the book is about beers, the probability that a vowel is followed by a vowel is probably greater than when it's about cats (if I'm right that e, a, o, u and i are the vowels, sorry I'm not from an english speaking country ;-)).

To get even better percentages, you could take a book yourself and calculate the percentages for that book. Then you just have to make sure that your sucker picks that book.

Peter

Mike Haven
09-15-2002, 09:13 PM
i don't think anyone has bothered their ass to count all the vowels and consonants in a book after markov did it, so you could well be right - in fact, he was a russian and he analysed a russian poem called eugene onegin, by pushkin, so maybe ...

the importance of a "markov chain" is that with some outcomes that appear to be random there is in fact a history that can be used to predict the next outcome, but that, given the history, only the immediately previous result is needed

take the weather in tel aviv - (another bet coming?) - a study showed that it is dry there (obviously) most of the time - if today is dry, there is a 75% chance that tomorrow is going to be dry too - but - if it is wet - there is a 66% chance that it will be wet tomorrow!

the study showed that over a period of 27 years there was no better prediction method for the next day's weather than by just knowing what that day's weather was!

there are almost bound to be markov chains in poker if we could only spot them - because a player has just won a bluff maybe he is more inclined to bluff the next hand? or not? maybe a player just having lost a big pot is more likely to win the next hand because he plays more aggressively? or because the big pot winner is paying less attention? or, vice versa, he wins again because he and everyone else perceives him as being on a roll? the next result may not be independent of the immediately previous result, but all other previous results may have no relevance at all

post script

during my research for this post i came upon an amusing short story at http://www.cs.concordia.ca/~faculty/grogono/starwar.html that you might care to read for enjoyment

Peter
09-16-2002, 10:50 AM
Great story, thanks for sharing.

Peter