Maths people, help!
In "Scarne on Cards", John Scarne discusses the odds for a game. He says this:
"The chances are 12220 to 9880 in their favour. [These numbers are definitely correct -- sil] That is, the percentage in their favour is 10-1/123."
Where's he getting that percentage from? How's he doing the calculation? I can't end up at that number, so I must be doing something wrong...
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Ok, our conclusion to this little puzzle is “Scarne did the calculation wrong”. The number is 2340/22100, which is an edge of about 10.58%, not “10-1/123” (which is about 10.081%).
This being an error is bolstered by further research: in his later Scarne’s Complete Guide to Gambling, he relates the same game (with a different story about it), lays out the same calculation, and comes up with an answer of 10 1/17% which isn’t right either!
Still, be tolerant: life is hard pre-calculators.
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it's not all that hard, though. Admittedly he's doing this in the context of writing a big long book, but didn't they have editors in the 50s? I -- no aficionado of long division -- just spent all of five minutes doing the calculation on paper and there it is, ~10.58%.
(I don't even know how you do this division to end up with a fraction rather than a decimal. Someone who was doing maths by hand in the fifties (and presumably learned to do so in the 1910s) will have to tell me (by ouija board).)
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@sil looks like around 10.6%?
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@simon see that's what I think! which is not 10-1/123. Hence not understanding.
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@sil I can't even understand the notation. is that "ten against one a hundred twenty third"? They seem to have moved the comma three places on one side and only two on the other?
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@mdione you know what I do. Here’s a picture of the relevant section from the book. My assumption was that it meant ten and one one-hundred-and-twenty-thirds, that is, (10 + (1/123)), or approx 10.081%. But maybe I’m reading it wrong!
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@sil or ten minus that? I'm not so savyy on math notation in English.
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@mdione it could mean that, but I think it's unlikely because then I'm really lost. The answer I get is about 10.58 which is at least ten point something; if the answer he's got is nine point something then we're way out in the woods :-)
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@sil I would calculate the margin as (12220 - 9880)/(12220 + 9880) or about 10.59%.
None of those numbers are divisible by 123, so presumably he's rounding. The margin is fairly close to 13/123: 0.1057 vs 0.1059. That corresponds to 10 + 70/123 as a percentage.
I wonder if he rounded at some earlier point in his calculations resulting in an increased error? I see the author is American, which probably explains why he doesn't just use decimals.
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@jamesh that’s the number I got too! I’ll post his whole calculation in a sec
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@sil Actually a more charitable reading is that the book is from 1949 and would have been written without the aid of a calculator. He may just have been working beyond the accuracy of whatever tools he was using.
I'm still not sure how expressing something as a fraction over 123 would increase someone's understanding of the result.
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@jamesh yeah. Although this is a simple game, so ought to be mathematically tractable. I am more dubious of his calculations of the bank’s edge at blackjack and so on which involve a lot more calculations! The book is excellently entertaining, though, filled with little stories about some mob gambler going down to the tune of a hundred grand in one night in a Havana hotel room and so on.
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@sil I'm not sure what "percentage in their favour" means in this context, but could we see a photo of the typesetting of "10-1/123"?
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@mavit here’s the whole page with full calculation (also for @jamesh )
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@sil @mavit @jamesh also this from call of the primes. either scarne and his editor are wrong or they are using some antiquated notation that none of us understand.
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@sil Incidentally, if I was emperor of the universe, I would ban percentages. They look like a number you could add, but are really a fraction you should multiply, and people find that confusing and difficult. You get people saying things like "The X party are ten percent ahead of the Y party in the polls", when 20% to %10 is actually double (that is, 100% ahead).
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@mavit yes! I have no intuition on these which means that I never properly understand what the number actually means!
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@sil 12220 / 9880 is 1.23(ish) which seems like that might be where it comes from, but may just be coincidental?
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@sil the odd thing is that he could have just said about 10% and that would have been fine. Why specify such a high level of precision?
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@sil TFW your Gell-Mann Amnesia is flaring up again.
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@slightlyoff what, me mistakenly trusting a gambling expert to get the maths right, or you mistakenly trusting me to get the long division right? :-)
the first two minutes were spent trying to divide 117 into 1105 instead of the other way around and being baffled as to why the answer began with a 9 when I knew in advance it ought to begin in a 1. Long time since I did any maths on paper :)
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@sil I haven’t thought about doing manual division in 30 years, I doubt I could anymore without a refresher.
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@daveshea the fickle hand of inaccurate historical reportage did not include the photo of the page where I got it wrong the first time. I think this is what people call "p-hacking"
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@sil I like to think there was a Douglas Adams style narrative where he died penniless due to an error in his book of lookup tables
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@simon possible, but I doubt it -- the bit in The Sting where Paul Newman does all the card shuffling stuff was actually him, so I suspect he had a bob or two tucked away, in addition to being very good at gambling which pays well if you get it right :)
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@sil You're dividing one integer by another - it's already a fraction! Just need to multiply by 100 to turn it into a percentage, then reduce by common factors.
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@sil So 234000/22100, 10 clearly goes, remainder is 13000/22100 = 130/221, which look coprime to me though I admit I can't be bothered to check.
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@sil which in fact reduces to 10/17, oops. Apparently not good at spotting factors of 13 in my head
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@cjwatson you don't know your 17 times table? tch the state of education these days, wasn't like that when people cared, etc, etc, etc
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@sil @cjwatson that's what Wolframalpha is for!
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@cjwatson sure, if you want it as a fraction you do the (long) division, get the remainder, put the remainder over the dividend, cancel. But then you get a fractional part which looks like something over 22100, which in no way is ever going to cancel to something over either 17 or 123, hence my puzzlement :)
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@cjwatson and since you've gotta basically do the long division anyway to get the integer part, why would you not carry on doing it to get a decimal part? I have polled a couple of Old People (to whom I happen to be related) and they said yeah, doing long division at school, you'd either report the answer as a decimal, or as 8 remainder 62 or whatever. Nobody seems to have even been taught a procedure which gives you an answer in the form 8 62/1317ths, or whatever. (I can see how to do it!)
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@cjwatson maybe it was different in the US, though (and pre-war, which my family aren't old enough to have been taught in :))
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@sil I sort of feel the transformation from "8 remainder 62 after division by 243" to "8 62/243" is obvious and just a matter of how you choose to spell it, though. But I agree the original author seems to have done something screwy, at a quick glance - almost as if they copied it down and mistranscribed the denominator or something.
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@cjwatson that’s fair comment, now that you say it like that, yeah!
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@sil my guess is that he might have been using a slide rule, but I'm not altogether certain how you'd do the "closest simple fraction" thing. Both incorrect fractions are "1/something" so maybe he was trying to calculate an inverse?
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@jamesh ooh slide rule. I didn’t think of a slide rule. Never used one :)
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@sil people would have hated doing long division back then too 😃
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@sil @jamesh slide ruler*?
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@mdione @sil One of these things: https://en.wikipedia.org/wiki/Slide_rule
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@jamesh @sil exactly. My father had one. He never taught me how to use it. We either lost it, broke it or both :(
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@jamesh @sil oh, they are called slide rule? Not ruler? Because my father's has a metallic edge with centimeters and millimeter etched into it to measure and draw straight line. In Spanish they're called "regla de cálculo", 'calculation ruler'.
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@sil Further to the guess about using slide rules, if he didn't properly understand log scale that could explain why his results skew downwards.
The half way point between two ticks on the rule is not 0.5, but about 0.3. That could explain error in the third decimal place (given the ticks would cover two decimal places).
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@sil Looking at this again, this answer is closer to making sense (while still being wrong.
22100 = 17 * 1300, so 17 as a denominator is plausible. In fact, the percentage is 10 + 10/17. Given it's just a dropped zero, he might have even had the right answer at some point.
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