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The Magic Cafe Forum Index » » Not very magical, still... » » The Most Counterintuitive Probability Paradox? (4 Likes) Printer Friendly Version

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S2000magician
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On Apr 19, 2019, Steven Keyl wrote:
That's it, Bill! You found it. Thanks, also, Jack for the wiki reference. I'm going to pick up Innumeracy. Looks like a great read. Thanks for that, too!

Much appreciated, gents.

My pleasure.

I just looked through my copy, and I was correct that that's where I found the reference on marrying.
Steven Keyl
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Jack's link led me to read more information on "optimal stopping theory" which has applications beyond mathematics into the realm of economics and finance. Might also, by extension, have practical application in politics when formulating policy. Wow... a lot to digest. Thanks again, Jack.

The book has been ordered, Bill. Looking forward to it.
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S2000magician
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On Apr 19, 2019, Steven Keyl wrote:
Jack's link led me to read more information on "optimal stopping theory" which has applications beyond mathematics into the realm of economics and finance. Might also, by extension, have practical application in politics when formulating policy. Wow... a lot to digest. Thanks again, Jack.

The book has been ordered, Bill. Looking forward to it.

On the topic of politics (which you mention), in his book Beyond Numeracy, Paulos has a fascinating discussion about voting schemes. You would find it fascinating, I have no doubt.
landmark
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In an organization I belong to, I just went through observing an election of nine candidates for three positions using Single Transfer Voting. What an ordeal the counting of the votes was!

I'll take the word of folks who I respect that there are many democratic advantages to this system over plurality takes all voting, but it certainly is far, far from intuitive if you take a look at the link.
R.S.
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On Apr 19, 2019, S2000magician wrote:
Quote:
On Apr 19, 2019, Steven Keyl wrote:
For everyone so inclined, I've got a mystery that needs solving...

The late Lew Brooks once told me about a method for determining the highest number of a group of numbers, the values of which are unknown at the outset. In other words, someone writes down, let's say, 10 numbers on 10 slips of paper. These can be any real numbers, and they do not need to be consecutive: 1, 10.5, 1,000,000, etc. The papers are mixed up and you start pulling out pieces of paper and reading the numbers on them. Somewhere between the 3rd and 7th piece you are able to stop and state which is the highest number in the series. If I recall, the success rate for this was ~60-70%, but I may not be remembering this correctly.

Does anyone know what I'm talking about? I've never heard about this anywhere else, but it may be a common problem/paradox in mathematics of which I'm unaware. I also don't remember the solution, but it was simple and straightforward. If anyone can point me in the proper direction, I'd appreciate it.

The procedure is this: discard (roughly) the first 37% of the numbers you see. (Don't forget what they are, because you'll need that; you simply remove them as candidates for being the highest.) As you continue looking at numbers, stop when you find one that's higher than any previous number: that's your choice for being the highest of the group.

The percentage that you discard is, as I say, roughly 37%. Exactly, it's 1/e, where e is the base of the natural logarithms: 2.718281828....

I don't recall the success rate of this procedure, but you're correct: it's surprisingly high.

In one of John Allen Paulos' books - Innumeracy I believe - the author uses this idea to formulate a plan for maximizing the probability that a person will marry the best person they can. He defines a heartthrob as someone who is better than everyone who has come before him or her. You start by estimating how many people you will meet in your lifetime whom you'd consider marrying. As you go through life you discard roughly the first 37% (i.e., i/e) of that estimated number, then marry the first heartthrob you meet after that group.

Voilà!


Thanks Bill! I also have "Innumeracy" (along with a couple other books by Paulos), and I highly recommend them! I'll need to revisit them, as they are very insightful.

Ron Smile
"It is error only, and not truth, that shrinks from inquiry." Thomas Paine
S2000magician
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Quote:
On Apr 20, 2019, R.S. wrote:
Quote:
On Apr 19, 2019, S2000magician wrote:
Quote:
On Apr 19, 2019, Steven Keyl wrote:
For everyone so inclined, I've got a mystery that needs solving...

The late Lew Brooks once told me about a method for determining the highest number of a group of numbers, the values of which are unknown at the outset. In other words, someone writes down, let's say, 10 numbers on 10 slips of paper. These can be any real numbers, and they do not need to be consecutive: 1, 10.5, 1,000,000, etc. The papers are mixed up and you start pulling out pieces of paper and reading the numbers on them. Somewhere between the 3rd and 7th piece you are able to stop and state which is the highest number in the series. If I recall, the success rate for this was ~60-70%, but I may not be remembering this correctly.

Does anyone know what I'm talking about? I've never heard about this anywhere else, but it may be a common problem/paradox in mathematics of which I'm unaware. I also don't remember the solution, but it was simple and straightforward. If anyone can point me in the proper direction, I'd appreciate it.

The procedure is this: discard (roughly) the first 37% of the numbers you see. (Don't forget what they are, because you'll need that; you simply remove them as candidates for being the highest.) As you continue looking at numbers, stop when you find one that's higher than any previous number: that's your choice for being the highest of the group.

The percentage that you discard is, as I say, roughly 37%. Exactly, it's 1/e, where e is the base of the natural logarithms: 2.718281828....

I don't recall the success rate of this procedure, but you're correct: it's surprisingly high.

In one of John Allen Paulos' books - Innumeracy I believe - the author uses this idea to formulate a plan for maximizing the probability that a person will marry the best person they can. He defines a heartthrob as someone who is better than everyone who has come before him or her. You start by estimating how many people you will meet in your lifetime whom you'd consider marrying. As you go through life you discard roughly the first 37% (i.e., i/e) of that estimated number, then marry the first heartthrob you meet after that group.

Voilà!

Thanks Bill! I also have "Innumeracy" (along with a couple other books by Paulos), and I highly recommend them! I'll need to revisit them, as they are very insightful.

Ron Smile

My pleasure, Ron.
LobowolfXXX
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Quote:
On Apr 8, 2019, landmark wrote:
You asked for it--it's turtles all the way down.

This has been discussed before by some really amazing folks here including Tomas B and Lobowolf XXX. There are many subthreads referred to as well. Screw your head on tight because it's about to be spinning:

https://www.themagiccafe.com/forums/view......orum=101


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Slim King
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I love this!!!!! Smile
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