statisticalrethinkin..

2.1. PROBABILITY IS JUST COUNTING 35occur. So they are just as honest as they are dependent upon the information used to definethem.2.1.4. More updating. But we’re not done. ere’s more data. So let’s see how the countsevolve as we fold in each remaining observation. e full sequence of globe tosses was:W L W W W L W L WYou’ve already accounted for the first two. I’ll go ahead and fill in the table, accounting foreach remaining observation in turn:proportion start W L W W W L W L W0.0 1 0 0 0 0 0 0 0 0 00.1 1 1 9 9 9 9 81 81 729 7290.2 1 2 16 32 64 128 1024 2048 16384 327680.3 1 3 21 63 189 567 3969 11907 83349 2500470.4 1 4 24 96 384 1536 9216 36864 221184 8847360.5 1 5 25 125 625 3125 15625 78125 390625 19531250.6 1 6 24 144 864 5184 20736 124416 497664 29859840.7 1 7 21 147 1029 7203 21609 151263 453789 31765230.8 1 8 16 128 1024 8192 16384 131072 262144 20971520.9 1 9 9 81 729 6561 6561 59049 59049 5314411.0 1 10 0 0 0 0 0 0 0 0is table is oriented with data along the top and the possible proportions on the le side.We started this analysis by saying there is at least one way for each possibility to be right, asseen in the “start” column. is just means that we’ll weight each possibility equally, for now.en the first W observation leads us to compute that there are more ways that the largerproportions could produce the data. at’s the first W column. As each subsequence observationis incorporated, the numbers of ways for each possibility to produce the sequence ofglobe tosses grows rapidly. By the time we reach the righthand edge, there are more than3-million ways for the proportion 0.7 to produce the observed sequence.2.1.5. From counts to probabilities. Of course the raw magnitude of these numbers is ofno interest. It’s only the relative magnitudes that matter. So usually we don’t work with theseraw counts. Instead we work with standardized counts. And that is all that’s le for us toarrive at probability in the usual form.Consider the numbers in the final column of the table in the previous section. If we addup all of these numbers, we get the total number of ways for all possibilities to produce thesequence of data. Let’s compute that number. Here’s R code to compute the values in thefinal column:ways_start