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tigerlily

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Hillary Barrett (Clarity) has asked me to draw members attention to the remarkable innovation featured at www.ichingmusic.com - this site is demonstrating a subtly devised formal system which has translated the 64 hexagrams of the I Ching into music in the form of piano duets (for yin and yang pianos) accompanying the splendid Vanyatrap Yogi Desh interpretations of the King Wen Texts.

This has to be the most lucid and accessible approach to the I Ching ever to be made available to Western enthusiasts. A delightful feature is the opportunity to send Divinations to friends for whom the reading may have specialk significance. The redaings are in the form of Videos which show the build-up of the hexagram(s), the Text and are accompanied by the music. Extraordinarily helpful and very clear.

I wish you all as much benefit and pleasure as I have enjoyed since discovering this wonderful new approach to the I Ching.

tigerlily
 

heylise

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Beautiful!!!
Except for one thing: when the coins spin, nines are conted as young yang, 6 as young yin. 7 as old yang, 8 as old yin.
So the primary hex. is okay, but the related one???

LiSe
 

hilary

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I did drop them a note via their contact form to say they had it backwards. Hope they have a tame programmer who can put it right!
 

heylise

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Three times the same question, three times the same answer!

Either they give the same answer all the time (checked, they don't) or they got a portable phone up to Shang Di and have a direct connection.

LiSe
 

jte

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Fun stuff - I never thought of Hex 9 as so... "jazzy" but definitely still enjoyed it...

:)

- Jeff
 

cal val

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I threw... having read LiSe's and Hilary's comments and knowing they've got old and new yin and yang reversed... and my resulting hexagram turned out to be very appropriate. I got 37 to 54. If I was being indulgent and not forward thinking, I would have been sad about the resulting hexagram 54... and their definition of it... Temptation Denied. Instead... knowing the Yi will answer me the best it can using the available resources, whether rigid rules are adhered to or not... I told the Yi later with coins that I was accepting the answer from the music site as valid, and their answer to that was hexagram 14.

Thank you for posting about the site, Tigerlily. I've been pondering the various meanings of 54 for a while now. Temptation denied is one I had not considered, and one that makes perfect sense in context. It's all making more sense to me every day.

Hilary, how exactly do they translate the hexagrams into music. I don't read music... just listen to it... *grin*... so I am clueless as to the mechanics that would be involved.

Love,

Val
 

argen

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Question on the coins,

I would like to know if I'm doing the rght thing,
The inscribed coin is a 3
The uninscribed coin is a 2

Please send me your answer on my questions

Thx.
Argen
 

hilary

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Hi Argen,

Sorry, I only just revisited this page and spotted your message. Either value for either side of the coin is fine, just so long as you're consistent (obviously!).

(The problem with the I Ching music site isn't to do with which side of the coin they use, but with them getting the numerical values wrong for the lines.)
 

nigel

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I have had the privilege of enjoying a most productive correspondence with Hilary (Clarity) on this subject, and, with her permission, I reproduce that correspondence here, since we both believe it will be of interest to members of the I CHING Community.

Dear Hilary Barrett,

A colleague has drawn my attention to the postings on your site relating to the Ritual of the Coins facility available at http://www.ichingmusic.com as discussed on your page http://www.onlineclarity.co.uk/I_Ching_community/messages/48/2468.html?1092758385.

May I offer a few words of clarification regarding this subject?

As many of your readers will be aware, different sources espouse different views with respect to which side of a Coin represents Yin, and which Yang. In my own experience, many I CHING adherents, East and West, use whatever coins are at hand - the distinction as to which side represents which aspect being entirely arbitrary.

When I embarked on the task of devising a formal system which would have the capability to render Hexagrams into harmonic progressions within the rubric of the Western Harmonic Tradition, I elected at a very early stage to employ as Texts the Vanyatrap Yogi Desh rendition, principally because I find his couplet format far and away the most 'musical'.

For the sake of consistency, I also decided to adhere to his interpretation of the Ritual of the Coins.

In any case, the critical consideration when consulting I CHING, surely, is to step aside from the minutiae of the nomenclature and immerse oneself in the Question and in contemplation of the Hexagrams and their accompanying Texts.

I hoped, and I believe (within limits) that my hopes have achieved some fulfilment, that by offering a means of consulting I CHING in a way that could (if desired) entirely dispense with the constraints of written and/or spoken language, I would be contributing to making the wisdom of I CHING more accessible and less shackled by pedantry. To this end, I would recommend to your readers that all that is necessary is to 'Formulate your Question and Spin the Coins' - the ensuing Hexagram(s), in the form of piano duets, will then make themselves manifest without further input from the supplicant. Each Hexagram can be replayed at will, and at the end of the sections on The Music and The Texts there is the opportunity to select any Hexagram and play it in isolation.

Yours respectfully

NW


Hilary very graciously replied as follows:


Dear Nigel,

Thanks for your email! I hope you also got the message I sent via your website about the coin-casting program at your site, and didn't have to find out about it via the posts at the I Ching Community. You'll notice everyone there was very appreciative of your site, and I've also heard from others who were delighted to discover it - one who had managed to use it to interest a friend in the I Ching for the first time.

I agree with you entirely that which side of the coin is counted as 2 or 3, yin or yang, is neither here nor there. In fact, not even Chinese authors are in agreement about it! (So I spend a lot of my time telling people not to worry about this.) The problem with the program at your site is something different. I hope you won't mind if I try to explain; please understand that I don't regard this mistake as detracting from the value of your musical work in any way!

You know that in casting a hexagram, each line can be either yin or yang, and either changing or unchanging. The correct values for each type of line are as follows:

unchanging yin: 8
changing yin: 6
unchanging yang: 7
changing yang: 9

I'm afraid that there is no disagreement between authors here: this is just how it is. The yarrow and three coin methods (whichever way anyone counts the sides of the coins) are just two different ways of arriving at the same numbers. So while your site's program is calculating the numerical value of each line quite correctly, it's then attributing the wrong kind of line to each number. It's labelling 7 as changing yin, 8 as changing yang, 6 as unchanging yin and 9 as unchanging yang.

This wouldn't matter, any more than which side of the coin is counted as 2 or 3 matters, if it weren't for the fact that the odds of getting a changing or unchanging line are different. Again with either yarrow or coin methods, your odds of getting a changing line are one in four. But since the ichingmusic site has the line values wrong, it gives odds of three out of four of getting a changing line. This makes for unnatural-feeling divinations... especially if anyone tries to look up and make sense of all those lines! (It's 'normal' to get just one or two changing lines in a reading.)

With best wishes,
Hilary


I reproduce my reply, below:


Dear Hilary,

Firstly, it is gratifying to note that we are in agreement with respect to the arbitrary nature of attributing Yin and Yang to the face (heads) and obverse (tails) sides of the coins.

Secondly, and with respect to the ?probability? aspect raised by some of your contributors, I believe I can best explain my decision by reference to the much-quoted Temesváry I Ging Wahrscheinlichkeits Tabelle - in English, I Ching Probability Table (Professor Stephan Temesváry,1975), which I reproduce below:

Erste Münze - Zweite Münze - Dritte Münze - Wert ? Wahrscheinlichkeit
(First Coin) - (Second Coin) - (Third Coin) - Value - Probability

Yang (3) + Yang (3) + Yang (3) = 9 = probability of 1/8

Yang (3) + Yang (3) + Yin (2) = 8
Yang (3) + Yin (2) + Yang (3) = 8 = } collective probability of 3/8
Yin (2) + Yang (3) + Yang (3) = 8

Yang (3) + Yin (2) + Yin (2) = 7
Yin (2) + Yang (3) + Yin (2) = 7 = } collective probability of 3/8
Yin (2) + Yin (2) + Yang (3) = 7

Yin (2) + Yin (2) + Yin (2) = 6 = probability of 1/8

This amounts to a simple statistical breakdown of the eight possible outcomes of spinning three coins, and is indisputably mathematically correct. However, as many scholars have observed, Professor Temesváry has overlooked an important practical aspect of I CHING divination; the Coins are not ordered. There is no first, second and third Coin.

This is easily forgiven; it is common practice to note down the outcome, and because of the linear nature of writing (or drawing symbols) an order manifests itself. But it is important to remember that this is a mere artefact of the notation process, and has no bearing on the Value (which takes no account whatever of any order). In this very important sense, there are only four possible outcomes to the coin-spinning procedure, which may be tabulated as follows:

GROUP OF 3 COINS - VALUE - PROBABILITY

3 Yang = 9 = probability of 1/4
2 Yang + 1 Yin (in no particular order) = 8 = probability of 1/4
2 Yin + 1 Yang (in no particular order) = 7 = probability of 1/4
3 Yin = 6 = probability of 1/4

It is more than thirty years since I travelled extensively in the East, but it was common then (as I am sure it is today) to spin all three coins simultaneously. The outcome was and is perceived in toto; only the process of notation imposes an order, which is in any case irrelevant, as is clearly apparent from the fact that the Value (and, therefore, the nature of the Line) is the same irrespective of the order of notation.

Finally, I believe you may be mistaken in thinking that there is no disagreement between sources regarding the Values and the types of Line which they signify. Many people throughout the East, scholars and laymen alike, subscribe to the view that (referring to the above Table) the correct interpretation is:

3 Yang Coins (9) conclusively represent an unchanging Yang Line;
2 Yang Coins plus 1 Yin Coin (8) represent a Line predominantly Yang, but tending (or moving) to Yin.
2 Yin Coins plus 1 Yang Coin (7) represent a Line predominantly Yin, but tending (or moving) to Yang, and
3 Yin Coins (6) conclusively represent an unchanging Yin Line.

This view possesses a certain logic, without being entirely compelling. Interestingly, Kwok Man Ho of the International Consultancy on Religion and Culture (ICOREC) in his 1986 book The Fortune Teller?s I Ching (co-authored by Dr Martin Palmer and Joanne O?Brien) espouses yet another view; 6 = moving Yin, 7 = Yin, 8 = Yang, and 9 = moving Yang. Opinions vary. As I mentioned in my earlier email, I elected to adhere to this interpretation in the interests of consistency. I am aware that there exists a deal of confusion relating to these details. I believe, however, that consistency within in any given rubric of beliefs can go some way towards minimizing such confusion ? ultimately these differences will be resolved as a matter of personal preference.

Most importantly, let me opine that in debating the merits and demerits of different interpretations of I CHING we are in danger of moving further away from the wisdom which it offers. It was in fact for this very reason that I committed myself to a musical rendition of I CHING ? so that we might have the option of experiencing it in a non-verbal way; there is no sophistry in music!

Notwithstanding our differing views, I am grateful to you for your most lucid explanation and kind words. Yours with kind regards,

NW

May I add to this correspondence my hopes that Community members will concur when I say, as I later said to Hilary, that having provided millions of people all over the world with inestimable guidance since King Wen was in short trousers (and before), I think we might agree that I CHING is possessed of sufficient fortitude to withstand a little wrangling over matters of procedure and interpretation - it is, after all, robust and well-balanced by definition.

Within many if not all of the great belief-systems, there exist variations in application. Some I CHING practitioners eschew both the yarrow stalks and the coins, preferring instead to draw one (or two) Hexagrams from a 64-card pack. Others employ the octahedral Tao-dice. Yet others merely riffle the pages of whatever translation they have to hand, stopping on impulse. Many (not I) have such exhaustive familiarity with I CHING as to disregard the subtleties of the individual Lines, and to immerse themselves directly in the unique ambience of the individual Hexagrams. This, too, can be profoundly instructive.

My objective in attempting a musical rendition of I CHING was to allow the ambience of each individual Hexagram to manifest itself in a form both familiar and comfortable to the Western ear. I accept that it will have its critics. Conversely, many have written to me to the effect that they have gained fresh insight from the experience.

Nigel Ward
http://www.ichingmusic.com
 
E

ewald

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Nigel,
I think that there is a simple way for you to find out what the probabilities are for the outcomes of throwing three coins: Just try it.

I threw a group of three coins, all together, 24 times. (The coins were not ordered.)
I got:

3 times - 3 heads
9 times - 1 heads and 2 tails
8 times - 2 heads and 1 tails
4 times - 3 tails.

Seems the probabilities are not 1/4 for each.
 

bradford_h

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Ewald-
You should get one of those little plastic cheat sheets on statistics.
And then throw your coins a hundred thousand more times.
 

hilary

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Actually Nigel himself put the correct 3 coin probabilities up there in the I Ging Wahrscheinlichkeitstabelle. If something (like the odds of 3 heads in a row) has odds of 1 in 8, it can't also have odds of 1 in 4 (ie 2 in 8). Not unless maths has changed a bit since I last hid behind the sofa to get away from it, anyway.
wink.gif


Dim memories of maths lessons: the odds of the coin coming up heads once is one half. Odds of this happening three times is 1/2 x 1/2 x 1/2. Which is 1/8. Though I probably shouldn't be trying this at home.

What interests/ baffles me, though, is Nigel's experience in the East of the line values being different. I'm sure the program on his site has the right probabilities for each numerical total, but it matches up line type and numerical value differently. Has anyone else (Brad?) been to China or read Chinese texts where moving lines are identified as 7s and 8s?

And if you think of moving lines as 7s and 8s, what do you do about the text that says '6 at the second place...' etc? Read the line texts only for lines that aren't moving? Hm, I suppose in that case it would all work out the same... but my head is spinning by now.
 

bradford_h

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Hi Hilary-
I've never seen a Chinese text where the Yao Ci are called 7's or 8's. It may be in some popular or marketplace version (oh those silly Xiao Ren), but certainly not in the Imperial Edition (Zhouyi Zhezhong), or in the Harvard Yencheng Zhouyi Yinde. It's always 6 & 9 as changing lines where scholarship matters.
Now, the value of heads and tails, or which side is which, varies, and is almost evenly split. But if these are two or three, then three of a kind can't add to 7 or 8. This was just started by some fool wanting to be an individual.
 
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ewald

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Brad - I'm sufficiently aware of how to calculate probabilities not to have a need for cheat sheets.
 

bradford_h

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Sorry Ewald-
Thought you were reinventing the wheels there.
The coin and stalk probabilities are such a frequent and boring topic on this and all other lists - they just need to be permanently posted somewhere so they don't need to come up again.
 

nigel

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Granted that one accepts the premiss that the order in which the coins fall is of no significance, it is self-evident that they must do so in one of four possible permutations;

3 Heads
2 Heads plus 1 Tail
1 Head plus 2 Tails
3 Tails

Anyone who doubts that each of these permutations carries a one-in-four probability of occurring might like to put the matter to the test empirically by conducting a simple though exceedingly tedious experiment.

First, construct a tetrahedral die. (A tetrahedron is a four-sided regular solid - each side being an equilateral triangle).

Then, using a red marker-pen and a blue one, mark the sides of the tetrahedron with coloured dots, in accordance with this colour plan:

1) Mark any side with 3 blue dots
2) Mark any other side with 2 blue dots and 1 red dot
3) Mark either of the remaining sides with 1 blue dot and 2 red dots
4) Mark the remaining side with 3 red dots

Now, roll the die a very large number of times (say, one hundred thousand), noting the results.

An analysis of these results will demonstrate that each side of the die comes up on something extremely close to one quarter of the total number of 'rolls', ie one in four.

Which is another way of saying that the probability for each of the four possible results is 1/4. QED.

Alternatively, if you subscribe to the view that the order of the coins does indeed have its significance, then the Temesváry Table is the one for you.

One other point: I have encountered every variation on the matter of which values represent the moving lines. Surely the salient fact here is that having once committed to a particular variation, there is a compelling case for sticking with it. Switching from one to another at will is effectively tantamount to choosing the Auxiliary Hexagram that best suits!

NW

http://www.ichingmusic.com
 

bradford_h

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Hi Nigel-
In case you want a second opinion, maybe you can locate this at a local library:
Gardner, Martin. ?Mathematical Games: The Combinatorial Basis of the I Ching, the Chinese Book of Divination and Wisdom.? Scientific American 230:1 (January, 1974): 109-13. Probabilities of divination methods.
 

heylise

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Math behind the sofa (Hi Hilary!)

Get 3 coins, and get your paint-box. Or get those 3 colors mailpolish you anyway don?t dare to put on your nails.
Paint each coin in a different color. E.g. Red, Green, Blue.
Now throw them.
Possibilities:
All coins yang is R+G+B+
R+G+B?
R+G-B+
R-G+B+
R+G-B-
R-G+B-
R-G-B+
R-G-B-

There are 8 different possibilities!!

All yang 1/8
One coin yin 3/8
One coin yang 3/8
All yin 1/8

Because they have different colors, you can see the difference. When they are identical, logical thinking makes the mistake to see the 3 throws with one yin as one single throw. It isn?t, it?s 3 throws.

LiSe
 

cal val

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LiSe...

What an idea... thank you! When I get home from work tonight, I'm going to paint my coins... rgb... just for the fun of it.

Love,

Val
 

bradford_h

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Hi LiSe-
That should have been clear enough.
Now, for those who followed you this far, it's a simple thing to substitute "bottom, middle and top line" for "red, green and blue", and what you get from your little matrix of possiilities is the full set of eight trigrams - father, three daughters, three sons and mother.
This has always been why I've preferred the coin probabilities to those of the stalks - it's so consistent with the Yi's own internal structure.
And I've long thought that the current Yarrow method is simply a mistake made by someone who never checked the math, even on the cursory level of testing suggested by Ewald. I doesn't matter whether they lived in the Zhou or the Song Dynasties. But I still suspect that this mistake was made in the Song and that the original stalk method is now long lost.
 

hilary

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<BLOCKQUOTE><HR SIZE=0><!-Quote-!><FONT SIZE=1>Quote:</FONT>

Granted that one accepts the premiss that the order in which the coins fall is of no significance, it is self-evident that they must do so in one of four possible permutations;

3 Heads
2 Heads plus 1 Tail
1 Head plus 2 Tails
3 Tails<!-/Quote-!><HR SIZE=0></BLOCKQUOTE>

Greetings from behind the sofa! Nigel, bless you, you are a lovely man and you've done the world a great service with your I Ching music, but you seem to be worse at maths than I am. Good grief.

The permutations?
3 heads
head + head + tail
head + tail + head
tail + head + head
3 tails
tail + tail + head
tail + head + tail
head + tail + tail

Which like LiSe said more colourfully, makes 8 possibilities. The point is any one of the three coins can come up heads to make a t+t+h combination. You've reckoned it as if it had to be just one particular coin of the 3. And like you say, it obviously makes no difference whether the coins are spun together or in succession. Doesn't matter how many coins are tossed at once, each one still has a 50:50 chance of coming up heads, right?

I think if you look at the odds in operation at your own site, you'll find that the numbers 9 and 6 come up about once in 8 times each.

P.S. Just kidding about the maths. You may be worse at statistics, but there's no way you could be worse at mental arithmetic.
wink.gif
 

hilary

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Another PS - sorry to go on and on. Please remember I have teachers in my ancestry, and make allowances...
blush.gif


Brad, has anyone ever invented a method that involved sorting yarrow stalks, but reproduced the coin odds?
 

bradford_h

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None that I've heard of. Plenty of attempts to reproduce the Stalk odds though.
But to even be considered as the method spoken about in the Xi Ci Zhuan it would have to use fifty stalks.
 

heylise

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Hilary, are you sure you did not invent fractals, behind that sofa? Because your connection of line-trigram is fantastic. Never thought of that.

LiSe
 

hilary

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That wasn't me, that was Brad. He has a much healthier relationship with sofas and statistics alike.
 

nigel

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Another 'spin' on the Coins debate:

I picked up in Hong Kong, I think it was in 1980, a beautiful boxed set of FOUR Coins. The accompanying literature specifies (in English, French, German and Dutch!) the following instructions:

Engraved side = Yin = 1
Plain side = Yang = 0

All four to be tossed simultaneously, with results valued as follows:

4 Heads = 4 = Yin
3 Heads + 1 Tail = 3 = Yin, moving to Yang
2 Heads + 2 Tails = 2 = perfectly balanced (throw again)
1 Head + 3 Tails = 1 = Yang, moving to Yin
4 Tails = 0 = Yang

Of course, the 'perfectly balanced' outcome (2)is in effect a 'void' throw, leaving the remaining four each carrying a probability of 1/4.

Apropos of which, Hilary, whilst your Maths is clearly better than you thought, you should keep your place behind the sofa warm, as I must (with grandmotherly kindness) take you to task about your English Comprehension! Specifically with regard to that passage of mine which you quoted;

"Granted that one accepts the premiss that the order in which the coins fall is of no significance, it is self-evident that they must do so in one of four possible permutations"

By listing the HHT, HTH, THH variants, you ARE placing the coins in an order, in fact in three different orders! You are confusing the means with the end.

Try this simple thought experiment:

a) You spin 3 Coins
b) You observe that the outcome has a value of 8
c) I ask you "How did you come that 8?"
d) You give one of the following replies:

"Heads, Heads, Tails", or
"Heads, Tails, Heads", or
"Tails, Heads, Heads"

But here lies your error. All three replies are in fact simply different DESCRIPTIONS of the same outcome - which is what I meant by (quote)"this is a mere ARTEFACT of the notation process"(unquote). There are not 3 different outcomes of value 8, only 3 different descriptions of the same outcome value 8. Ergo, since the same applies to the outcome 7, it is clear that I am quite correct in saying that there are only four different outcomes, each with a probability of 1/4. The tetrahedral die demonstrates this conclusively.

Let us agree to differ. Before posting this entry, I drew;

Hexagram 61 - Chung Fu - Childlike Honesty
Preferring fish to pig, he travels by the coast road.
The pig, in his turn, takes no offence.

http://www.ichingmusic.com/iching/html/theoracle/choose_speed.html?61-61
 

hilary

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Oh Nigel... there are at least three people here setting me a beautiful example of how to deal with this with maturity and grace by just letting it lie. Either they are just more advanced human beings than I am, or they have fewer teachers in the family. Or maybe - probably - both.

Anyway, if I don't have one more go at explaining this, I shall go stark mad. (If you'd received 62 rather than 61, you'd have had more chance of peace!) So please forgive my graceless and 'childlike' honesty! I will try to make this my last attempt.

One thing you could do is just to re-read LiSe's post with great care, and maybe borrow some nail polish from somewhere and try the experiment she suggests. She explained this a whole lot better than I did, so after reading that you may be able to skip all that follows. (Or use 3 different coin values - probably less embarrassing than getting the nail polish
mischief.gif
)

But I can't quite leave it at that, so I'll have one more go.

We have two 'kinds' of event here:
- the possible outcome
- the ways you could get this outcome

There are 4 outcomes: 6,7,8,9. Or you could say the possible outcomes were 3 heads, 3 tails, 2 heads and a tail or 2 tails and a head. Either way, definitely 4 outcomes.

But in some situations, one outcome is more likely than another. In other words, there are more chances for that outcome to happen... or more different ways it could happen.

For instance... suppose I have a bag with marbles in. Some are red, some are black. If I pull one out at random, it'll be either red or black. Definitely, 2 outcomes. But if I try to persuade you that there's a 50:50 chance that the next marble I pull out will be red, will you believe me?

You might want to count the marbles first. If I've got 3 red marbles and 1 black one - well, you can probably see the odds straight away. But bear with me while I go through it...

4 marbles in the bag in total. 2 colours. That's 2 possible outcomes if I pull out one marble, but 4 different marbles I could actually have picked: 4 different ways to reach that outcome.

3 of the ways are red marbles, one of the ways is a black marble. 3 times in 4, on average, I'll get a red marble. Next time I pull one out, I have a 3:4 chance of a red one. 1:4 chance of a black one.

If you have one coin, and toss it, there are two outcomes. H or T. And there is also just one head and one tail on your coin: one way for each outcome to happen. So that's 2 things that can happen (outcomes), and 1 chance of each happening. 1:2 chance of h or t.

Making a marble bag to represent these same odds would be pretty straightforward. One black marble to represent the 'heads' side of the coin, one red marble to represent the 'tails' side. Equal chance of tossing a head or a tail; equal chance of pulling out either marble.

Pick up a second coin, and this will get more complicated, though. You have three possible outcomes: tt, ht, hh. Does that mean that each outcome has an equal chance of happening? Maybe, but you can't take it for granted - like you wouldn't take it for granted that there was a 50:50 chance of a red or black marble from that bag of mine until you'd counted the marbles. Before you know how likely each outcome is, you need to count up the number of ways it could happen.

Let's say (thank you, LiSe) that we have a 1p and a 2p coin here. I think you'll agree that this doesn't make any difference to the results? Each coin still has a 50:50 chance of landing h or t, no matter what size it is.

There are 3 different possible outcomes here, we're agreed. tt, ht or hh. The marble bag for this one will need an extra colour to represent the third possible outcome.

But how many different things could actually happen when you toss the coins?

1p tails, 2p tails
1p heads, 2p heads
1p heads, 2p tails
1p tails, 2p heads

4.

That's like having 4 marbles in a bag: if you pull one out at random, it could be any one of the four. Each colour for a marble is a possible outcome. Each marble is a possible way to get that outcome.

So... making a marble bag to represent our 2 coin experiment. We have 3 possible outcomes: tt, ht, hh, so we represent each of these with a colour. Say tt=red, hh=black, ht=green. But we have four different ways the two coins could land, so we need one extra marble. Another green one.

1 coin needs 2 marbles in the bag.
2 coins need 4.

If you agree with me thus far, then 3 coins will be a doddle. If you don't, I'm clean out of ideas... at least for now. And well off topic and quite tedious, sorry Brad.
 

bradford_h

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A young man known to my daughter owns a fairly nice, but twelve year old pickup truck.
A couple of years ago, a malfuntion of the transmission left hm without a reverse gear. Not being especially well off, but being possessed of a circular driveway at home, he postponed getting this problem fixed. These days he only goes places which don't require him to back up. It's a little like living with a moderate handicap - if you've lost one eye you can still see. You can still walk, even if you limp on one leg. And only every now and then does he get into an embarrassing situation. But you can't stop him from going forward.
I think this may be a little like that.
 

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