statisticalrethinkin..

2.1. PROBABILITY IS JUST COUNTING 33reverse the order, because the problem is symmetric: if there are 2 out of 10 ways to see W,then there are 8 out of 10 ways to see L.2.1.3. Update the possibilities. We’re not really interested directly in the numbers of waysthat each possibility can produce the data. ose counts are just a means to an end. Instead,we’re directly interested in the number of ways that each possibility could be correct. So nowwe use both the initial relative counts for each possibility in combination with the numbersof ways to see the data to produce a new count of ways that each possible proportion of watercould happen. ese counts are, as before, proportional to the relative plausibility of eachpossibility, because things that can happen in more ways are more plausible.e way to update the counts for each possible proportion is just to count. Consider thefirst possibility, a proportion of zero. ere was initially 1 (relative) way for it to be correct,by assumption. For this one way, we found that there are zero ways for it to produce theobservation W. So there are 1 × 0 = 0 ways for a proportion of zero to be correct. Similarlyfor the second possibility, there was 1 way for 0.1 to be correct, and we counted 1 way (outof every 10 ways) for it to produce the observation W, resulting in a new count of 1 × 1 = 1.And so on, for all of the other possibilities. e new counts, and therefore new relativeplausibilities, are shown in the table.proportion 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0ways 1 1 1 1 1 1 1 1 1 1 1aer seeing W 0 1 2 3 4 5 6 7 8 9 10at’s easy enough. But consider now the second observation in the sequence of globetosses, L. e numbers of ways for each possible proportion to produce this observationare given by the righthand plot in FIGURE 2.2, as explained before. To update again thenumbers of ways each possible proportion could be correct, again take the current numbersand multiply them by the respective numbers of ways each possibility could produce thedata. is gives us:proportion 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0ways 1 1 1 1 1 1 1 1 1 1 1aer seeing W 0 1 2 3 4 5 6 7 8 9 10ways to see L 10 9 8 7 6 5 4 3 2 1 0aer seeing WL 0 9 16 21 24 25 24 21 16 9 0You can confirm these calculations quickly in R. First make a list of initial ways for eachpossible proportion to be correct:ways