AN
ANALYSIS OF THE SHOOTING-ROOM PARADOX.
Doug Clark, 22nd November 2003. mailto:[email protected]
Last revised 14th December 2003.
Introduction.
A description of the paradox, as presented by Bostrom (1998), is given below:):
“In the shooting-room experiment we are to imagine a room of infinite capacity. First a batch of ten people are led into this room. A pair of dice is thrown in front of their eyes. If a double six comes up they are all shot. Otherwise they leave the room safely and a new batch, this one containing a hundred people, is thrust in. The process continues, with each consecutive batch ten times larger than the previous one, until there is a double six; whereupon the people in the room at that time are shot and the experiment ends.
Suppose you have been thrust into the room. You are asked to estimate the odds of leaving safely. One the one hand, since whether you will leave or not will be determined by the throw of a fair pair of dice, it seems that you have a 35/36 chance of exiting alive. On the other hand, 90% of all people who are in your situation will be shot, so it seems you have only a 10% chance of exiting alive. That is the paradox.”
John Leslie, as reported by Bostrom (1998), thinks that “there is a radical difference for what the person in the shooting room should believe depending on whether the random mechanism (the two dice) are deterministic or not.” I do not believe that the shooting-room experiment could distinguish between indeterminate (truly random) and pseudo-random dice, however it is of interest to see how the probability of exiting alive may depend upon the ‘random mechanism.’ In the analysis below two potentially random mechanisms, the dice and the selection of subjects, are identified.
Analysis
Each stage in the experiment, when a new batch of people enter the room, can be numbered from 1 to n, where the number of people in the batch is 10n at stage n.
The probability of not leaving safely, P(S = E), is determined by two parameters; the number of the stage at which the experiment ends (S) and the number of the stage at which you enter the room (E), where E £ S. The proportion of participants who die at the end of the experiment (D) is a function of S:
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D = |
10S |
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l = S |
. |
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S 10l |
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l = 1 |
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As S increases, D ® 0.90.
D = 1 for S = 1
0.91 for S = 2
0.90 for S > 2.
Four scenarios are possible, corresponding to whether E and S are constants or random variables. In all cases D is as defined above.
1) Both
parameters are constant.
Let S = s and E = e. Since E £ S, P(s = eêe £ s) = 1 for e = s
0 for e < s.
2) S is a random variable and
E is a constant.
P(S = n) = (1/36).(35/36)(n – 1).
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Let E = e. Since e £ S, P(S = eê e £ S) = |
(1/36).(35/36)(e – 1) |
= |
1 |
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(35/36)(e – 1) |
36 |
. |
3) S is a constant and E is a random variable.
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P(E = nê E £ S) = |
10n |
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l = S |
. |
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S 10l |
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l = 1 |
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Let S = s, P(s = Eê E £ s) = |
10s |
= D |
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l = s |
. |
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S 10l |
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l = 1 |
4) Both S and E are random variables.
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P((S = n) AND (E = m)êE £ S) = (1/36).(35/36)(n – 1). [ |
10m |
] |
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l = n |
. |
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S 10l |
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l = 1 |
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Therefore:
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P(S = EêE £ S) = |
n = ¥ |
[ (1/36).(35/36)(n – 1).[ |
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] ] = 0.90. |
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S |
10n |
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n = 1 |
l = n |
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S 10l |
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l = 1 |
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Comments.
(1) This analysis models the shooting-room experiment using random variables and a relative frequency interpretation of probability theory. In the protocol of the experiment the participants are powerless to alter the outcome. Therefore the experiment could be repeated with different participants, one of which could be designated as ‘you’, without altering the probability distribution of the outcome. The experiment could also be done using balls instead of people with one ball marked as ‘you’ and the probability distribution of the outcome would remain the same. The fact that the experiment has the potential to continue indefinitely does not preclude analysis in terms of relative frequencies in repeated trials, since as n increases the probability of the experiment having stopped converges to 1.
(2) Leslie, as reported by Bostrom (1998), has claimed that if the dice are random then P(S =E|E £ S) = 1/36 whereas if the dice are deterministic then “Disaster is what will come to over 90 per cent of those who will ever have been in your situation.” Eckhardt is said to believe that P(S =E|E £ S) = 1/36 in both scenarios (Bostrom, 1998.) Although the protocol specifies that the dice are fair it does not say whether the procedure for choice of participants is random or deterministic, so it is not clear which of these scenarios Leslie and Eckhardt are referring to.
The above analysis of the shooting-room experiment supports Leslie’s thesis that the value of P(S =E|E £ S) is dependent on whether parameters are constants (deterministic) or random variables, although it disagrees with him on details. If the choice of participants in the experiment is taken to be deterministic then scenarios 1 and 2 correspond to Leslie’s deterministic and random dice. In scenario 1; P(s =e|e £ s) = 1 or 0, rather than 0.9 as claimed by Leslie although D = 0.9 (for n > 2). For a completely deterministic scenario it would seem that probability defined by relative frequency could only be equal to 1 or 0. In scenario 2; P(S =e|e £ S) = 1/36, in agreement with Leslie; however there is no paradox, since D is a proportion rather than a probability. If the choice of participants in the experiment is taken to be random then for scenario 3 in which the dice are deterministic, P(s =E|E £ s) = D and for scenario 4 in which both parameters are random variables, P(S =E|E £ S) = 0.9. These results are not in agreement with Leslie.
(3) If a parameter has been set as a random variable this need
not imply that values of the parameter are generated by a random process; a pseudo-random process could generate the
same distribution, provided that the
initial conditions of the experiment were allowed to vary in an appropriate
fashion for repeated trials of the experiment.
In practice there is a limitation on how precisely the initial
conditions of the experiment can be set.
If it was possible to have identical conditions for each trial and
external influences were unable to influence the course of the experiment, then
it would be possible to distinguish a truly random from a pseudo-random
process.
4) In conclusion; the shooting-room paradox can be modelled using random variables, the protocol is ambiguous with regard to the mechanism for selection of participants and the value of P(S =E|E £ S) is dependent upon whether S and E are set as constants or random variables.
References.
Bostrom (1998.) The Doomsday Argument; a Literature Review. http://www.anthropic-principle.com/preprints/lit/
