{"id":1822,"date":"2014-10-23T12:42:43","date_gmt":"2014-10-23T12:42:43","guid":{"rendered":"https:\/\/skepticink.com\/reasonablyfaithless\/?p=1822"},"modified":"2014-10-23T12:42:43","modified_gmt":"2014-10-23T12:42:43","slug":"coins-marbles-and-the-resurrection","status":"publish","type":"post","link":"https:\/\/skepticink.com\/reasonablyfaithless\/2014\/10\/23\/coins-marbles-and-the-resurrection\/","title":{"rendered":"Coins, marbles and the resurrection"},"content":{"rendered":"

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My previous post, How (not) to argue about the resurrection<\/a>, was the first in a series on the Minimal Facts argument for the resurrection of Jesus. \u00a0This is a quick follow-up post – please read the original post in order to get up to speed.<\/p>\n

In the previous post, I addressed the following\u00a0statement\u00a0made by William Lane Craig:<\/p>\n

In order to explain that the resurrection is improbable, [a skeptic]\u00a0needs not only to tear down all the evidence for the resurrection, but he needs to erect a positive case of his own in favour of some naturalistic alternatives. \u00a0(William Lane Craig, debate with Bart Ehrman, 2006 \u2013 transcript<\/a>)<\/p><\/blockquote>\n

I\u00a0argued that it is not true that in order\u00a0to demonstrate that a certain event E is unlikely, one must argue in favour of some alternative event F. \u00a0This would be to ask someone to show that\u00a0the probability, P(E), of E occurring is less than\u00a0the probability, P(F), of F occurring. \u00a0Sure, if one could comprehensively demonstrate that P(E)<P(F), then we would have a quick proof that E is unlikely (ie, has a probability of less than 50%). \u00a0But to show that P(E)<50%, we don’t need to show that P(E)<P(F) for some single alternative event F; we need to show that P(E)<P(anything but E). \u00a0(In other words, while showing that P(E)<P(F) for some F is\u00a0sufficient <\/em>to show that E is unlikely, it is not\u00a0necessary\u00a0<\/em>in general. \u00a0This is also completely leaving aside the fact\u00a0that it is up to the proponent of the argument to argue in favour of E being likely, not merely to challenge others to prove E is unlikely.) \u00a0And I gave a couple of examples to back up this point.<\/p>\n

Well, a couple of people have objected\u00a0that those examples I gave (please read the previous post for the details) were different from the case of the resurrection in a crucial way – that all the alternative hypotheses had equal probabilities. \u00a0It is certainly the case that the alternatives in the stories I told are all equally likely. \u00a0And it is certainly the case that the various possible explanations for the proposed “minimal facts” related to the resurrection stories (claims of an empty tomb, post-mortem experiences, etc) are not equally probable. \u00a0For example, I would say that P(stolen body) > P(Jesus didn’t really die) > P(Jesus didn’t even exist) > P(aliens were involved) > P(Zeus was involved).<\/p>\n

However, I think that the equal probabilities of the alternatives in the examples I gave\u00a0are nothing more than an unfortunate distraction from the point I was making: that in order to demonstrate an event E is unlikely, one is not required to compare the “E occurred” hypothesis to a single alternative “F occurred” hypothesis – rather, one must compare the “E occurred” hypothesis to the “something other than E occurred” hypothesis. \u00a0But, just to be sure, this post is simply to give a third story that makes the same point, but is not open to the same objection. \u00a0So here we go…<\/p>\n

I decide to play a third guessing game with my students. \u00a0This time I ask Carla\u00a0to go into another room where she finds\u00a0a huge barrel filled with thousands of marbles each labelled with one of the numbers\u00a01,2,3,4,5,6,7,8,9,0. \u00a0There is also a coin on a\u00a0table, and a set of instructions that read as follows:<\/p>\n

Step 0. \u00a0Toss the coin 20 times. \u00a0If you get 20 consecutive tails, go to Step 1. \u00a0Otherwise, pick 20\u00a0marbles at random\u00a0and write the sequence of\u00a0\u00a0labels on the\u00a0blackboard in the order you\u00a0draw them from the barrel, then return to class.<\/p>\n

Step 1. \u00a0Toss the coin again. \u00a0If you get heads, write 0\u00a0on the board and return to class. \u00a0If you get tails, go to the next step.<\/p>\n

Step 2. \u00a0Toss the coin again. \u00a0If you get heads, write 00 on the board and return to class. \u00a0If you get tails, go to the next step.<\/p>\n

Step 3. \u00a0Toss the coin again. \u00a0If you get heads, write 000 on the board and return to class. \u00a0If you get tails, go to the next step.<\/p>\n

…<\/p>\n

Step 19. \u00a0Toss the coin again. \u00a0If you get heads, write 0000000000000000000 (19 zeros) on the board. \u00a0If you get tails,\u00a0write \u221e\u00a0(infinity)\u00a0on the board. \u00a0In either case, return to class.<\/p>\n

When Carla\u00a0returns, I announce to the class \u201cI claim that Carla\u00a0wrote 0 on the board\u2033. \u00a0Am I likely to be right?<\/p>\n

So we know that Carla has written some number sequence on the board. \u00a0She has written either (i) a 20 digit number sequence, or (ii) a sequence of zeros of length at most 19, or (iii) the infinity symbol, \u221e. \u00a0As it happens, we can work out the exact probability of any of these possible sequences being on the board. \u00a0Let’s do this for each possible sequence:<\/p>\n

(i) \u00a0In order for Carla to have written some fixed 20 digit number sequence, X, on the board, she must have first not flipped 20 tails, and then randomly chosen the marbles to produce sequence X. \u00a0Multiplying the probabilities for each of these yields (2^20-1)\/2^20 \u00d7 1\/10^20, or approximately 0.0000000000000000001%.<\/p>\n

(ii) \u00a0In order for Carla to have written some fixed sequence 000……0 of N zeros on the board (where 1 \u2264\u00a0N \u2264\u00a019), she must have first flipped 20 tails (to get up to Step 1), and then another N-1 tails (to get up to Step N), and then a head on her next flip. \u00a0So the probability that she has written this sequence\u00a0is 1\/2^(20+N).<\/p>\n

(iii) \u00a0Finally, in order for Carla to have written \u221e\u00a0on the board, she must have flipped 39 tails in a row. \u00a0So the probability that she has written\u00a0\u221e is 1\/2^39, or approximately 0.00000000018%.<\/p>\n

The probabilities may be\u00a0listed in decreasing order as follows:<\/p>\n