In two previous posts I discussed Martin Smith's paper "Why throwing 92 heads in a row is not surprising" [1].
I argued that the all-heads sequence is more surprising than a more balanced one (composed of roughly equal numbers of heads and tails), although the two have the same probability of occurrence based on the prior information (fair and independent coins), because the first event challenges this information, while the second does not.
Prompted by an email exchange with Martin (whom I thank again for his patient and detailed replies!) I would like to discuss here cases where I believe no particular outcomes would be surprising. Let us take an example from the same paper, the lottery. I agree with the author that a draw consisting of consecutive numbers (e.g. 123456) is not surprising, nor is any other pattern or apparently random sequence, which all have the same probability of occurrence.
In my view, this is simply because –for the lottery case– equiprobability assumption is just an uninformative prior. Finding a patterned sequence does not challenge our conviction, because there is nothing to challenge: any outcome will do equally well. For the coin toss, on the other hand, the equiprobability of all sequences stems from the very strong conviction that the coins are (1) unbiased and (2) independent, and the all-head outcome does challenge it.
1. Martin Smith, Why throwing 92 heads in a row is not surprising, Philosophers' Imprint (forthcoming) 2017.↩
I argued that the all-heads sequence is more surprising than a more balanced one (composed of roughly equal numbers of heads and tails), although the two have the same probability of occurrence based on the prior information (fair and independent coins), because the first event challenges this information, while the second does not.
Prompted by an email exchange with Martin (whom I thank again for his patient and detailed replies!) I would like to discuss here cases where I believe no particular outcomes would be surprising. Let us take an example from the same paper, the lottery. I agree with the author that a draw consisting of consecutive numbers (e.g. 123456) is not surprising, nor is any other pattern or apparently random sequence, which all have the same probability of occurrence.
In my view, this is simply because –for the lottery case– equiprobability assumption is just an uninformative prior. Finding a patterned sequence does not challenge our conviction, because there is nothing to challenge: any outcome will do equally well. For the coin toss, on the other hand, the equiprobability of all sequences stems from the very strong conviction that the coins are (1) unbiased and (2) independent, and the all-head outcome does challenge it.
1. Martin Smith, Why throwing 92 heads in a row is not surprising, Philosophers' Imprint (forthcoming) 2017.↩
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