Ancestors

Written by Stuart Langridge on 2025-01-04 at 09:35

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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Written by James Henstridge on 2025-01-04 at 10:40

@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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Toot

Written by Stuart Langridge on 2025-01-04 at 10:46

@jamesh that’s the number I got too! I’ll post his whole calculation in a sec

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Descendants

Written by James Henstridge on 2025-01-04 at 10:58

@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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Written by Stuart Langridge on 2025-01-04 at 11:00

@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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