• Coins, marbles and the resurrection

     

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

    In the previous post, I addressed the following statement made by William Lane Craig:

    In order to explain that the resurrection is improbable, [a skeptic] needs 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.  (William Lane Craig, debate with Bart Ehrman, 2006 – transcript)

    I argued that it is not true that in order to demonstrate that a certain event E is unlikely, one must argue in favour of some alternative event F.  This would be to ask someone to show that the probability, P(E), of E occurring is less than the probability, P(F), of F occurring.  Sure, 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%).  But 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).  (In other words, while showing that P(E)<P(F) for some F is sufficient to show that E is unlikely, it is not necessary in general.  This is also completely leaving aside the fact that 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.)  And I gave a couple of examples to back up this point.

    Well, a couple of people have objected that 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.  It is certainly the case that the alternatives in the stories I told are all equally likely.  And 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.  For 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).

    However, I think that the equal probabilities of the alternatives in the examples I gave are 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.  But, 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.  So here we go…

    I decide to play a third guessing game with my students.  This time I ask Carla to go into another room where she finds a huge barrel filled with thousands of marbles each labelled with one of the numbers 1,2,3,4,5,6,7,8,9,0.  There is also a coin on a table, and a set of instructions that read as follows:

    Step 0.  Toss the coin 20 times.  If you get 20 consecutive tails, go to Step 1.  Otherwise, pick 20 marbles at random and write the sequence of  labels on the blackboard in the order you draw them from the barrel, then return to class.

    Step 1.  Toss the coin again.  If you get heads, write 0 on the board and return to class.  If you get tails, go to the next step.

    Step 2.  Toss the coin again.  If you get heads, write 00 on the board and return to class.  If you get tails, go to the next step.

    Step 3.  Toss the coin again.  If you get heads, write 000 on the board and return to class.  If you get tails, go to the next step.

    Step 19.  Toss the coin again.  If you get heads, write 0000000000000000000 (19 zeros) on the board.  If you get tails, write ∞ (infinity) on the board.  In either case, return to class.

    When Carla returns, I announce to the class “I claim that Carla wrote 0 on the board″.  Am I likely to be right?

    So we know that Carla has written some number sequence on the board.  She 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, ∞.  As it happens, we can work out the exact probability of any of these possible sequences being on the board.  Let’s do this for each possible sequence:

    (i)  In 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.  Multiplying the probabilities for each of these yields (2^20-1)/2^20 × 1/10^20, or approximately 0.0000000000000000001%.

    (ii)  In order for Carla to have written some fixed sequence 000……0 of N zeros on the board (where 1 ≤ N ≤ 19), 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.  So the probability that she has written this sequence is 1/2^(20+N).

    (iii)  Finally, in order for Carla to have written ∞ on the board, she must have flipped 39 tails in a row.  So the probability that she has written ∞ is 1/2^39, or approximately 0.00000000018%.

    The probabilities may be listed in decreasing order as follows:

    • P(0) = 1/2^21 ≈ 0.000048%
    • P(00) = 1/2^22 ≈ 0.000024%
    • P(000) = 1/2^23 ≈ 0.000012%
    • P(0000) = 1/2^24 ≈ 0.0000060%
    • …………
    • P(0000000000000000000) = 1/2^39 ≈ 0.00000000018%
    • P(∞) = 1/2^39 ≈ 0.00000000018% (same as previous)
    • P(X) = (2^20-1)/2^20 × 1/10^20 ≈ 0.0000000000000000001%, where X is any fixed 20 digit number sequence.

    So there are a whole lot of possible number sequences that could be on the board.  And, as you can see, the single most likely sequence is 0, a single zero (the very sequence I predicted).  But it would be madness to suggest I was probably right.  The probability that I am right is just 0.000048%, worse than one in two million.

    But again, the students couldn’t possibly “erect a positive case” in favour of any particular alternative.  The reason for this is simply that P(X) is minuscule for any alternative sequence X.  But I am still almost certainly wrong.  This is not because some other sequence is more likely than 0 (we know this is not the case).  It is because the combined total of the likelihoods of all the other sequences adds up to more than 50% – in fact, the exact figure is 1 – 1/2^21 ≈ 99.99995%, almost 100%.  (Interestingly, even though the 20 digit number sequences have the lowest individual chances of occurring, the probability that some 20 digit number sequence is on the board is 1 – 1/2^20 ≈ 99.9999%, again almost 100%.  Even though each such sequence is extraordinarily unlikely, it is almost certain that one of them is written on the board.)  So, of course, the students would be warranted to think I was probably wrong, even though (a) my prediction is more likely than any other specific prediction, and (b) they couldn’t possibly “erect a positive case” in favour of any other specific prediction.

    And the same could well be true in the case of the resurrection claim.  Without actually calculating the exact probabilities of the various possible explanations of the proposed “minimal facts” related to the resurrection (or at least giving plausible estimates), it is simply not enough for the apologist to claim (or even prove) that the resurrection hypothesis is more probable than any other specific alternative hypothesis.  It is conceivable (though highly unlikely, in my opinion) that the resurrection hypothesis is more probable than any other specific alternative hypothesis, while still being exceedingly unlikely.

    Again, I want to stress that I am not implying this story is numerically equivalent (in either the number of possible cases, or the probabilities of the various cases) to the details of the resurrection story.   But this story makes exactly the same point as before, while also avoiding the objection about the alternatives having equal probabilities.  It should also make it quite clear that by tweaking the details (perhaps by involving dice, playing cards, Rubik’s cubes, etc), one could come up with similar stories with just about any probability distribution one wanted.  (I was tempted to add an extra detail that would have made the 0 sequence much more likely than any of the others (00, 000, etc), but relented for the sake of simplicity.  Please feel free to ask for the details if you aren’t sure how this could be done.)

    Another nice feature of the current story is that there are a huge number of sequences with extraordinarily tiny probabilities (if one wished, one could tweak the details to ensure these tiny probabilities were all different), and only a small handful of sequences with comparably bigger probabilities.  I suspect that a believer in the resurrection would consider this to be similar (but not necessarily numerically equivalent) to the situation with the resurrection – they would probably consider that there are loads of extremely unlikely explanations (aliens, Zeus, and any number of implausible naturalistic stories – one is only limited by one’s imagination) and only relatively few more plausible (but still very unlikely, the believer might think) stories such as the “stolen body”, “apparent death” or “conspiracy” theories, for example.

    But in the end, the point remains the same.  To explain why it is unlikely that 0 is written on the board, one does not need to prove that some other sequence is more likely to be on the board.  We shouldn’t be comparing the “sequence 0 is on the board” hypothesis to any particular “sequence X is on the board” hypothesis.  We should be comparing the “sequence 0” hypothesis to the “anything other than sequence 0” hypothesis.

    And the same is true of the resurrection.  As I said at the end of the previous post:

    The burden of proof is on the apologist.  The apologist should not demand a skeptic prove some specific “non-resurrection” hypothesis.  Rather, the apologist needs to show that the “resurrection” hypothesis is more probable than the “anything but the resurrection” hypothesis.  In my opinion, no apologist has ever succeeded in doing this, and this series will outline my reasons for coming to this conclusion.

     

    Category: AtheismChristianityFeaturedHistoryJesusResurrectionTheismWilliam Lane Craig

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    Article by: Reasonably Faithless

    Mathematician and former Christian
    • Tony_Lloyd

      (T)o 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).

      This is the nub.
      P(F) is not P(E), but P(F) is not P(not-E). This settles the matter.

      Perhaps an easier and quicker, though less mathematical, explanation is that were we
      required to give a rival explanation before rejecting an explanation we would
      have to accept that Uri Gellar had magic powers (his explanation) if we couldn’t
      explain how he did his trick.