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We know that Dogmeat won on three occassions on which it rained, and there were four rainy days in total.So Dogmeat's probability of winning, given that it is now raining, is 3 / 4, or 0.75, or 75%, however you like to write it.This is because three of his previous five wins have been on rainy days.
Therefore, all other things being equal (which they're not — see below), the probability of Dogmeat winning the next race can be estimated from his previous wins: 5 / 12, or 0.417, or 41.7%, however you prefer to express it.
Fleetfoot, on the other hand, appears to be a better bet at 58.3%.
So, given only the information that we have, and everything else being equal (including the odds offered by the bookmakers on these two horses) you'd want to back Fleetfoot.
A 58.3% likelihood of winning is more favourable than a 41.7% likelihood. It turns out that on three of Dogmeat's previous five wins, it had rained heavily before the race.
Note that the additional information — that Dogmeat won three times out of four on a raining day — shifts his probability of winning this current race from 41.7%, to 75%.
More importantly, it means that now Dogmeat is more likely to win than Fleetfoot, even though Fleetfoot has won more races overall.In this article I will explain how Bayes' theorem works, from first principles.I assume of the reader no knowledge of mathematics beyond elementary arithemetic.To do this, we need to examine four possible situations: Dogmeat wins when it rains; Dogmeat wins when it doesn't rain; Dogmeat loses when it rains; Dogmeat loses when it doesn't rain.We know that Dogmeat achieved three of his five wins on rainy days, and it only rained once when he lost. Since there were twelve races, the number of times he lost on a sunny day must be 12 - (3 1 2), which is 6.In a celebrated court case (R v Adams  1 Cr App R 377, for any lawyers that are interested) Lord Bingham, one of the UK's most senior judges, refused to allow the defence to present an argument to the jury based on Bayes' theorem.He conceded that it was a methodologically sound approach, but that it would 'confuse the jury'. It has been put forward as a solution to a number of important problems in, among other disciplines, law and medicine.These problems are concerned with such matters as determining the likelihood that a particular suspect committed a murder if his fingerprints are found on the weapon, or the likelihood that a person who tests positive for HIV really has an HIV infection. However, most people can't see that there are any difficulties here at all.However, it had rained only once on any of the days that he lost.It appears, therefore, that Dogmeat is a horse who likes 'soft going', as the bookies say. Now, how does this extra information affect where you should put your money?