tag:blogger.com,1999:blog-264226589944705290.post8642053391023644991..comments2022-11-19T01:08:31.131-08:00Comments on God Plays Dice: The probability that 901 coins have total value $100Michael Lugohttp://www.blogger.com/profile/15671307315028242949noreply@blogger.comBlogger3125tag:blogger.com,1999:blog-264226589944705290.post-91375656606311086882010-05-20T20:23:49.943-07:002010-05-20T20:23:49.943-07:00The exact answer isn't hard to compute if we c...The exact answer isn't hard to compute if we consider each coin to be equally likely - it's a classic example of generating functions. I'll give an example and link to Wolfram alpha for the math, but to do so I have to use a smaller problem ($50 instead of $100). <br /><br />The number of ways one could make $50 using 901 coins (.01, .05, .10, .25) is simply the coefficient in front of the x^5000 term in the expansion of:<br /><br />(x+x^5+x^10+x^25)^901<br /><br /><a href="http://www.wolframalpha.com/input/?i=Coefficient%5B(x%2Bx%5E5%2Bx%5E10%2Bx%5E25)%5E901,x,5000%5D" rel="nofollow"> link </a><br /><br />Which is ~ 10^475<br /><br />If you can choose those coins any way you like you get:<br /><br />Eval -> (x=1) (x+x^5+x^10+x^25)^901<br /><br />Which is ~ 10^542<br /><br />Thus the odds are about 10^(-67), very, very small. This is of course a different answer then posted as we assumed each coin is equally likely (and I used $50 since Wolfram craps out at $100). Also as Efrique said, this _particular_ event is rare - however we typically look at the probability of getting some _range_, ie. the probability of 901 coins being < $100.Hookedhttps://www.blogger.com/profile/13293647191115859617noreply@blogger.comtag:blogger.com,1999:blog-264226589944705290.post-20610613652370535912010-05-01T23:04:46.367-07:002010-05-01T23:04:46.367-07:00The problem with this sort of calculation is that ...The problem with this sort of calculation is that the event of interest is defined <i>after</i> the observation, not before.<br /><br />The effective "event" is therefore <br /> ... whatever would make the person say "What are the odds of THAT!??" <br /><br />The chances that *something* odd would happen is not all that low.<br /><br />There are many combinations of large numbers of coins that would make one say "what are the odds of that?". The restriction to 901 coins seems entirely arbitrary, since if it had been 899 coins, would that not also have generated a similar question? The restriction to $100 also seems arbitrary - might not a similar question have been generated with a total of some other very round amount?<br /><br /><br />When there are lots of replications, it becomes even less surprising that something might happen...<br /><br />If someone wins the lottery, they might ask "wow, what are the odds of <i>that</i>", but almost anyone else winning it would ask the same question... and what are the chances that <i>somebody</i> won the lottery?<br /><br />Any calculation of the probability of a post-specified event as if it were a pre-specified event is ridiculously wrong, and leads to many of the abuses of statistics we see. People have ended up in jail on the basis of bad calculations like this.Efriquehttps://www.blogger.com/profile/08526031804261484547noreply@blogger.comtag:blogger.com,1999:blog-264226589944705290.post-22222662868050152982010-05-01T13:30:21.301-07:002010-05-01T13:30:21.301-07:00It's harder to get if we exclude copper... but...It's harder to get if we exclude copper... but perhaps more interestingAnonymousnoreply@blogger.com