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Subject: Fun Fact: The 504 Paradox rss

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Steven Mitchell
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How many times would a gaming group need to play a random setup of 504 to have a 50/50 chance of playing the same game twice?

Spoiler (click to reveal)
27 times.


How many times before they have a 99.9% chance of playing the same game twice?

Spoiler (click to reveal)
84 times.


Inspired by the famous Birthday Paradox.
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Kevin Peters Unrau
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Fun!

I guessed only 3 high on the first question but a full 31 high on the second. Not exactly intuitive numbers are they? (At least for a non-math person like me.)
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Luis Fernandez
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LOL
 
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Joseph Anderson
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I agree with your first figure but for your second one I got a different result:
Spoiler (click to reveal)
69 was 99.2%; 80 was 99.9%


and to be clear this is the chance of having played one world twice after that many games, not the chance that that game is a duplicate game.
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Michael Alexander
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For those interested in why this is, see the Birthday Problem.
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Steven Mitchell
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knaves wrote:
I agree with your first figure but for your second one I got a different result:


On second calculation, ha! it seems I was looking at 98.9%. The result of 99.9% was actually...

Spoiler (click to reveal)
84 games. Or 82 games, if you're rounding the hundredths place up.


Original post amended.
 
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Russ Williams
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patton1138 wrote:
How many times would a gaming group need to play a random setup of 504 to have a 50/50 chance of playing the same game twice?


Next math challenge:

How many times would a gaming group need to play a random setup of 504 to have a 50/50 chance of having played every possible world at least once?
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Tim Puls
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Haven't played the game so far. Thus the question. Is game 123 the same as game 321 and 132 and so forth?
 
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Russ Williams
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wredo wrote:
Haven't played the game so far. Thus the question. Is game 123 the same as game 321 and 132 and so forth?

As noted on the game's BGG entry ("More Information"):

Note that 504 = 9 * 8 * 7 = the number of distinct permutations of 3 items from a set of 9. The order of the 3 game modules is significant.
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Ethan Nicholas
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russ wrote:
How many times would a gaming group need to play a random setup of 504 to have a 50/50 chance of having played every possible world at least once?


I'm not smart enough to do the math, but computer simulation suggests it is a bit over 3,300 plays.
 
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Chad Urso McDaniel
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What's the paradox?
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Aaron Morgan
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chadm wrote:
What's the paradox?


The "Birthday Problem" is often referred to as such because the answer seems to defy logic.
 
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Steven Mitchell
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chadm wrote:
What's the paradox?


It's what's known as a veridical paradox: the correct answer completely flies in the face of intuition and seems absurdly wrong, but can actually be proven to be correct. Most people guess the correct answer is around half the overall set of possibilities — in this case, 252.

The other way you could state it is: 'Imagine your game group has played 504 XX number of times. What is the probability that they have played the same game twice?' Most people will guess something far below the actual answer.

(Of course, if you already are familiar with the underlying math, you've also essentially undermined common intuition, so it won't seem so nearly paradoxical.)
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