<!DOCTYPE html><HTML lang="en"> <head><meta charset="utf-8"> <title>Smith (Martin) - Why Throwing 92 Heads in a Row Is Not Surprising (Theo Todman's Book Collection - Paper Abstracts) </title> <link href="../../TheosStyle.css" rel="stylesheet" type="text/css"><link rel="shortcut icon" href="../../TT_ICO.png" /></head> <BODY> <CENTER> <div id="header"><HR><h1>Theo Todman's Web Page - Paper Abstracts</h1><HR></div><A name="Top"></A> <TABLE class = "Bridge" WIDTH=950> <tr><th><A HREF = "../../PaperSummaries/PaperSummary_22/PaperSummary_22835.htm">Why Throwing 92 Heads in a Row Is Not Surprising</A></th></tr> <tr><th><A HREF = "../../Authors/S/Author_Smith (Martin).htm">Smith (Martin)</a></th></tr> <tr><th>Source: Philosophers' Imprint, Vol. 17, No. 21, October 2017, pp. 1-8</th></tr> <tr><th>Paper - Abstract</th></tr> </TABLE> </CENTER> <P><CENTER><TABLE class = "Bridge" WIDTH=600><tr><td><A HREF = "../../PaperSummaries/PaperSummary_22/PaperSummary_22835.htm">Paper Summary</A></td><td><A HREF = "../../PaperSummaries/PaperSummary_22/PapersToNotes_22835.htm">Notes Citing this Paper</A></td><td><A HREF = "../../Notes/Notes_12/Notes_1257.htm">Link to Latest Write-Up Note</A></td></tr></TABLE></CENTER><hr><p><B>Authors Citing this Paper</B>: <A HREF = "../../Authors/S/Author_Smith (Martin).htm">Smith (Martin)</A></p></P> <hr><P><FONT COLOR = "0000FF"><U>Author s Abstract</U><FONT COLOR = "800080"><ol type="1"><li>Tom Stoppard s "Rosencrantz and Guildenstern Are Dead" opens with a puzzling scene in which the title characters are betting on coin throws and observe a seemingly astonishing run of 92 heads in a row. Guildenstern grows uneasy and proposes a number of unsettling explanations for what is occurring. Then, in a sudden change of heart, he appears to suggest that there is nothing surprising about what they are witnessing, and nothing that needs any explanation. He says  & each individual coin spun individually is as likely to come down heads as tails and therefore should cause no surprise each individual time it does. </li><li>In this article I argue that Guildenstern is right  there is nothing surprising about throwing 92 heads in a row. I go on to consider the relationship between surprise, probability and belief. </li></ol></FONT><hr><FONT COLOR = "0000FF"><B>Comment: </B><ul type="disc"><li>See <a name="W5757W"></a><A HREF = "https://quod.lib.umich.edu/p/phimp/3521354.0017.021/1" TARGET = "_top">Link</A>.</li><li>I m writing <a name="1"></a><A HREF="../../Notes/Notes_12/Notes_1257.htm">this note</A><SUP>1</SUP> as a rebuttal / analysis of this paper.</li><li>Annotated printout filed with <a name="8"></a>"<A HREF = "../../BookSummaries/BookSummary_06/BookPaperAbstracts/BookPaperAbstracts_6527.htm">Hains (Brigid) & Hains (Paul) - Aeon: Q-S</A>" for want of a better home. </li></ul><BR><FONT COLOR = "0000FF"><hr><br><B><u><U><A HREF="#On-Page_Link_P22835_2">Write-up</A></U><SUB>2</SUB><a name="On-Page_Return_P22835_2"></A></u> (as at 02/08/2018 15:48:58): Surprising Coin-Toss Sequences (and Bridge Hands)</B><BR><br><u>Introduction / Motivation</u> <ul type="disc"><li>This paper is a review of <a name="3"></a>"<A HREF = "../../Abstracts/Abstract_22/Abstract_22835.htm">Smith (Martin) - Why Throwing 92 Heads in a Row Is Not Surprising</A>". </li><li>The above paper claims that there s nothing  surprising about a fair coin coming up heads 92 times in a row, basically on the grounds that any random string HTTHHHTTTHTHT ... is equally probable. </li><li>While I agree with the latter claim, I don t agree with conclusion drawn from it. In fact, it s preposterous, and this is the sort of philosophy paper  and it won a prize  that brings  philosophy of X into contempt amongst practitioners of X  in this case mathematics, or statistics  not that I count myself as one of those. </li><li>There are faint resonances with ancient discussions I ve had with friends on the question of the probabilities of unusual events  eg of miracles  where I know our views differed. This is a less contentious case, I think. Here, the probability of the event is agreed upon and has a calculable numerical value, which is not the case with the probabilities of miracles. However, it is relevant to their perceived credibility. </li><li>it s important when responding to a  preposterous argument to get to the bottom of it and find out exactly what s wrong with it. My suspicions are that  irrespective of psychology and how our brains might be wired  we need to invoke things like the <U><A HREF="#On-Page_Link_P22835_3">central limit theorem</A></U><SUB>3</SUB><a name="On-Page_Return_P22835_3"></A>, <U><A HREF="#On-Page_Link_P22835_4">information theory</A></U><SUB>4</SUB><a name="On-Page_Return_P22835_4"></A>, analogies with the <a name="2"></a><A HREF="../../Notes/Notes_11/Notes_1172.htm">Sorites</A><SUP>5</SUP> Paradox, the <U><A HREF="#On-Page_Link_P22835_6">Lottery Paradox</A></U><SUB>6</SUB><a name="On-Page_Return_P22835_6"></A> and the like.</li><li>The author (<a name="7"></a><A HREF = "../../Authors/S/Author_Smith (Martin).htm">Martin Smith</A>) has written lots of (fairly contentious) stuff on testimony versus probabilities, which I ve collected and am looking forward to studying. </li></ul><BR><u>Detailed Arguments</u><BR><ul type="disc"><li><b>Examples of  Surprising Events</b><BR>&rarr; The light not coming on when I flick the switch<BR>&rarr; A colleague promising to attend a meeting and  no showing <BR>&rarr; My car is not where I left it</li><li>Smith <b>aims to argue</b>  in contrast  that throwing 92 heads would <u><b>not</b></u> be surprising. </li><li>The above is a <b>normative</b> claim  we might well be surprised, but  Smith will argue  we ought not to be. <BR>&rarr; This claim  it substantiated  has far-reaching consequences for what we should believe, given our limited evidence, in other circumstances. </li><li>The <b>Conjunction Principle</b>: what is the surprisingness-rating of (e<sub>1</sub> & e<sub>2</sub>) given the surprisingness-ratings of the individual events? <ol type="i"><li>The surprisingness of the conjunction two unsurprising events would also be unsurprising  Smith claims  if the two events are unconnected (like two successive tosses of a fair coin). </li><li>If the conjunction of the two events were impossible, then the surprisingness of the conjunction occurring would be extreme. </li><li>So, Smith claims this <b>conjunction principle</b>: <FONT COLOR = "800080">If it s unsurprising for event e<sub>1</sub> to happen, and it s unsurprising for event e<sub>2</sub> to happen, and these two events are independent of one another, then it s unsurprising for e<sub>1</sub> and e<sub>2</sub> to both happen. </FONT> </li><li>Smith now posits that the <b>conjunction principle</b> can be iterated so that 92 consecutive Hs are not surprising, given that no individual H is surprising. </li><li>Two earlier attempts to define the <b>conjunction principle</b> are addressed:-<BR>&rarr; <U><A HREF="#On-Page_Link_P22835_7">George Shackle</A></U><SUB>7</SUB><a name="On-Page_Return_P22835_7"></A> (1950s-60s): has, for two independent events, Sup (e<sub>1</sub> & e<sub>2</sub>) = max {Sup (e<sub>1</sub>), Sup (e<sub>2</sub>)}, where Sup in [0 , 1]. <BR>&rarr; <U><A HREF="#On-Page_Link_P22835_8">Wolfgang Spohn</A></U><SUB>8</SUB><a name="On-Page_Return_P22835_8"></A> (1980s): has, for two independent events, Sup (e<sub>1</sub> & e<sub>2</sub>) = sum {Sup (e<sub>1</sub>), Sup (e<sub>2</sub>)}, where Sup in Z+. <BR>For both Shackle s  mathematical theory of surprise and Spohn s  ranking theory , a completely unsurprising event  like a single coin-toss resulting in H  has Sup of 0, so the conjunction of two such events  and indeed any number thereof  also has Sup of 0, ie. is completely unsurprising. </li></ol></li><li><u><b>Surprise versus Unlikeliness</b></u> <ol type="i"><li>But, isn t 92 consecutive Hs rather <em>unlikely</em>? Yes, its probability is 2 x 10<sup>-27</sup>; near miraculous, and  many think  very surprising.</li><li>Smith quotes some injudicious remarks by a <U><A HREF="#On-Page_Link_P22835_9">trio of mathematical greats</A></U><SUB>9</SUB><a name="On-Page_Return_P22835_9"></A>: d Alembert in 1760s, Cournot in the 1840s and Borel in 1942 to the effect that such low-probability events <em>never happen</em> ( Borel s Law ). Smith asks whether such a claim  while an exaggeration  might be approximately true  such events very rarely happen, and are therefore surprising.</li><li>Smith thinks the claim  far from being near the truth  is almost the exact opposite. His argument is that we can have a situation where <u>every</u> outcome is highly <em>improbable</em> but <u>not</u> one where <u>every</u> outcome is <em>surprising</em>. This shows that surprise and probability come apart. The reason for this is that in the coin-tossing case, <em>any</em> outcome is just as unlikely as any other; so, HTTHHTHT & is just as unlikely as HHHHHHHH & . For 92 coin tosses, <em>each</em> outcome  one of which is <em>bound</em> to happen  is a 1-in-5,000-trillion-trillion event; so  when a 1-in-5,000-trillion-trillion event <em>does</em> happen, it should not be surprising. </li><li>Smith tries to reinforce this by appeals to other improbable but unsurprising events  like the precise temporal, volumetric and molecular dimensions of a breath; these  in their exact measurements  are even more unlikely than the coin-toss sequences. Similarly, the precise time my phone rings. We don t want to be in a perpetual state of surprise. A vague claim  a phone-call  this weekend  can be likely, but a specific one will be unlikely. </li><li>Smith sums up by saying that the conjunction principle has allowed him to prove that 92 Hs is not surprising, even though it is extremely unlikely. </li></ol></li><li><u><b>Expectation</b></u> <ol type="i"><li>Smith admits that 92 consecutive Hs are not to be <em>expected</em> where expectation is used in the mathematical sense of the probability weighted average of the possible values of a random variable. He notes that the probabilities of the number of heads in 92 tosses approximates to a normal distribution, with the bell curve peaking at 46 and 92 Hs more than 9 standard deviations from the mean. Smith asks whether this is one of the occasions where an extreme divergence from the mean ought to elicit surprise.</li><li>Smith has an interesting argument against the tempting conclusion that surprise is appropriate. He notes that there s a 73.8% chance that there will be between 40 and 50 Hs out of 92 tosses. He makes an analogy between this and the claim that  my phone will ring over the weekend . The reason either claim is so likely is that there are so many  individually improbable  ways for the claim to be satisfied. </li><li>So, why is this not a conclusive argument in favour of surprise  very mild in this case  being appropriate if expectation isn t met? </li><li> </li></ol></li><li></li><li> </li></ul><BR><BR><u>Analysis</u><ul type="disc"><li>This argument doesn t just apply to coin-tossing. I quote a passage from <a name="9"></a>"<A HREF = "../../BookSummaries/BookSummary_02/BookPaperAbstracts/BookPaperAbstracts_2730.htm">Kelsey (Hugh) & Glauert (Michael) - Bridge Odds for Practical Players</A>":- <ol type="1"><FONT COLOR = "800080"><b>Four Complete Suits</b><ul type="square"><li>The question that naturally springs to mind at this stage is, what are the chances of all four players being dealt a complete suit? Well, there are 4! or 24 ways in which each player can receive a complete suit, and division by 24 <U><A HREF="#On-Page_Link_P22835_10">leaves us with</A></U><SUB>10</SUB><a name="On-Page_Return_P22835_10"></A> odds of <U><A HREF="#On-Page_Link_P22835_11">2,235,197,406,895,366,368,301,559,999</A></U><SUB>11</SUB><a name="On-Page_Return_P22835_11"></A> to 1 against. If the entire adult population of the world were to play bridge in every waking moment for ten million years, it would still be ten million to one against one of these perfect deals turning up. </li><li>So how can we account for all the newspaper reports of four players in a bridge game each receiving a complete suit? The answer is invariably a joker, not in the pack but amongst the players or, more probably, in the ranks of the kibitzers. It is not too hard to switch a pack without being spotted.</FONT> </li></ul></ol> </li></ul><BR><HR><BR><U><B>In-Page Footnotes</U></B><a name="On-Page_Link_P22835_2"></A><BR><BR><U><A HREF="#On-Page_Return_P22835_2"><B>Footnote 2</A></B></U>: <ul type="disc"><li>This is the write-up as it was when this Abstract was last output, with text as at the timestamp indicated (02/08/2018 15:48:58). </li><li><A HREF = "../../Notes/Notes_12/Notes_1257.htm">Link to Latest Write-Up Note</A>. </li></ul><a name="On-Page_Link_P22835_3"></A><U><A HREF="#On-Page_Return_P22835_3"><B>Footnote 3</A></B></U>: <ul type="disc"><li>For the Central Limit Theorem, see Wikipedia: <a name="W5808W"></a><A HREF = "https://en.wikipedia.org/wiki/Central_limit_theorem" TARGET = "_top">Link</A>.</li><li>See also the Law of Large Numbers; Wikipedia: <a name="W5810W"></a><A HREF = "https://en.wikipedia.org/wiki/Law_of_large_numbers" TARGET = "_top">Link</A>. </li></ul> <a name="On-Page_Link_P22835_4"></A><U><A HREF="#On-Page_Return_P22835_4"><B>Footnote 4</A></B></U>: See <a name="4"></a>"<A HREF = "../../Abstracts/Abstract_22/Abstract_22589.htm">Goodman (Rob) & Soni (Jimmy) - The bit bomb</A>". <a name="On-Page_Link_P22835_6"></A><BR><BR><U><A HREF="#On-Page_Return_P22835_6"><B>Footnote 6</A></B></U>: <ul type="disc"><li>Smith has written on the Lottery Paradox; ie:-<BR>&rarr; <a name="5"></a>"<A HREF = "../../Abstracts/Abstract_22/Abstract_22868.htm">Smith (Martin) - A Generalised Lottery Paradox for Infinite Probability Spaces</A>" and<BR>&rarr; <a name="6"></a>"<A HREF = "../../Abstracts/Abstract_22/Abstract_22864.htm">Ebert (Philip A.), Smith (Martin) & Durbach (Ian) - Lottery Judgments: A Philosophical and Experimental Study</A>"</li><li>See Wikipedia: <a name="W5809W"></a><A HREF = "https://en.wikipedia.org/wiki/Lottery_paradox" TARGET = "_top">Link</A> </li></ul> <a name="On-Page_Link_P22835_7"></A><U><A HREF="#On-Page_Return_P22835_7"><B>Footnote 7</A></B></U>: <ul type="disc"><li>For George Shackle , see Wikipedia: <a name="W5800W"></a><A HREF = "https://en.wikipedia.org/wiki/G._L._S._Shackle" TARGET = "_top">Link</A></li><li>Smith references <FONT COLOR = "800080"><em>Decision Order and Time in Human Affairs</em>, 2nd ed. (Cambridge University Press, 1969) [Contains Shackle s most detailed presentation of his ideas about surprise. His axioms, and his struggles over axiom 7, can be found in chapter X.]</FONT> </li><li>Wikipedia makes no mention of this book, unfortunately. </li></ul> <a name="On-Page_Link_P22835_8"></A><U><A HREF="#On-Page_Return_P22835_8"><B>Footnote 8</A></B></U>: <ul type="disc"><li>For Wolfgang Spohn , see Wikipedia: <a name="W5801W"></a><A HREF = "https://en.wikipedia.org/wiki/Wolfgang_Spohn" TARGET = "_top">Link</A> </li><li>Smith references <FONT COLOR = "800080"><em>The Laws of Belief</em> (Oxford University Press, 2012) [Spohn s definitive presentation of ranking theory and its various applications. Discusses Shackle and surprise in section 11.1. The law of conjunction for negative ranks is principle 5.16 in chapter 5.]</FONT> </li><li>This book is referenced by Wikipedia, and Spohn s work looks worth following up! </li></ul> <a name="On-Page_Link_P22835_9"></A><U><A HREF="#On-Page_Return_P22835_9"><B>Footnote 9</A></B></U>: <ul type="disc"><li>For these three, see Wikipedia:-<BR>&rarr; d Alembert: <a name="W5805W"></a><A HREF = "https://en.wikipedia.org/wiki/Jean_le_Rond_d%27Alembert" TARGET = "_top">Link</A>,<BR>&rarr; Cournot: <a name="W5806W"></a><A HREF = "https://en.wikipedia.org/wiki/Antoine_Augustin_Cournot" TARGET = "_top">Link</A> <BR>&rarr; Borel: <a name="W5807W"></a><A HREF = "https://en.wikipedia.org/wiki/%C3%89mile_Borel" TARGET = "_top">Link</A>. </li></ul> <a name="On-Page_Link_P22835_10"></A><U><A HREF="#On-Page_Return_P22835_10"><B>Footnote 10</A></B></U>: <ul type="disc"><li>The total number of deals had just been calculated as 52! / (13!)<sup>4</sup>.</li><li>Ie. 53,644,737,765,488,792,839,237,440,000 </li></ul> <a name="On-Page_Link_P22835_11"></A><U><A HREF="#On-Page_Return_P22835_11"><B>Footnote 11</A></B></U>: <ul type="disc"><li>5 x 10<sup>-28</sup></li><li>This is of the same order, more or less, as the probability of our 92 consecutive Hs. </li></ul> <FONT COLOR = "0000FF"><HR></P><a name="ColourConventions"></a><p><b>Text Colour Conventions (see <A HREF="../../Notes/Notes_10/Notes_1025.htm">disclaimer</a>)</b></p><OL TYPE="1"><LI><FONT COLOR = "0000FF">Blue</FONT>: Text by me; &copy; Theo Todman, 2018</li><LI><FONT COLOR = "800080">Mauve</FONT>: Text by correspondent(s) or other author(s); &copy; the author(s)</li></OL> <BR><HR><BR><CENTER> <TABLE class = "Bridge" WIDTH=950> <TR><TD WIDTH="30%">&copy; Theo Todman, June 2007 - August 2018.</TD> <TD WIDTH="40%">Please address any comments on this page to <A HREF="mailto:theo@theotodman.com">theo@theotodman.com</A>.</TD> <TD WIDTH="30%">File output: <time datetime="2018-08-16T09:29" pubdate>16/08/2018 09:29:47</time> <br><A HREF="../../Notes/Notes_10/Notes_1010.htm">Website Maintenance Dashboard</A></TD></TR> <TD WIDTH="30%"><A HREF="#Top">Return to Top of this Page</A></TD> <TD WIDTH="40%"><A HREF="../../Notes/Notes_11/Notes_1140.htm">Return to Theo Todman's Philosophy Page</A></TD> <TD WIDTH="30%"><A HREF="../../index.htm">Return to Theo Todman's Home Page</A></TD> </TR></TABLE></CENTER><HR> </BODY> </HTML>