Honors Core Beyes

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Michael Daniel Oct 2 2006 Sober, pg. 200 – 206 Bayes’s Theorem Thesis: In order to make a good hypothesis about the probabilities involved in any situation or event we need to have both prior possibilities and observations. Bayes’s theorem was presented to the Royal Society of London and is now accepted as fact. It shows that we need to use probability to understand evidence and confirmation. Before an observation is made we need to formulate a hypothesis H. We will call the evidence from the ob
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  Michael DanielOct 2 2006Sober, pg. 200 – 206 Bayes’s TheoremThesis: In order to make a good hypothesis about the probabilities involved inany situation or event we need to have both prior possibilities and observations.Bayes’s theorem was presented to the Royal Society of London and is nowaccepted as fact. It shows that we need to use probability to understand evidence andconfirmation.Before an observation is made we need to formulate a hypothesis H. We will callthe evidence from the observation O. When we combine H with O we obtain the probability that H is true. This is represented by the expression pr(H).To illustrate this principle the book answers the question, “What is the probabilitythat a card drawn at random from a standard deck is a heart, given that it is red?” This iswritten pr(A|B)=(pr(A&B)/pr(B). The probability that the card is red is 100% because itis a given observation. 100% = 1 so B =1. 50% of all red cards in a standard poker deck are red. 50% = 0.5 so A = 0.5. This means that the statement works out to (1*.5)/1 = 0.5.0.5 = 50% so the probability that any card you have drawn is a heart, given that it is red is50%. Another way to write this is pr(A|B) = 50%.The book then says that the reverse of the above example is true. In other words, pr(B|A) = (pr(A&B))/pr(A). I don’t understand this because if this is supposed to still bea reverse of the above statement then pr(B|A) means, “What is the probability that a carddrawn at random from a standard deck is a red card, given that it is a heart?” When we plug the numbers in we get (1*0.5)/1 = 0.5 or 50%. All hearts are red unless you are not playing with a standard deck or are color blind. The probability is 100% that your card is  red if it is a heart. This formula sets the probability at 50%. I am not certain exactlywhere the problem is. Either the book, the formula, or my understanding is wrong.Beyes theorem states that, given initial hypothesis pr(H) and the conditional probability after the observation pr(H|O), pr(H|O) = (pr(O|H)pr(H))/(pr(O). In other words, you take the product of the probability suggested by your observation and the probability suggested by your hypothesis and divide it by the probability suggested byyour observation. There are three things that observation O can possibly do to H. First, itcan confirm hypothesis H, second, it can disconfirm hypothesis H and third it can beirrelevant to H.If pr(O|H) = 1 then we have achieved unity. In the event of unity, observationfails to confirm. In other words, if there is a 100% chance of something happening because your hypothesis is deduced then observation can not confirm or disconfirm thehypothesis. True predictions fail to confirm using observation.If we want to get the probability of one hypothesis given another hypothesis wemultiply. In other words if H1 = 1% and H2 = 50% we will have the expression pr(H1|H2). The example given in the book is that we have an urn that has a 1% chance of beingfilled with nothing but green balls and a 99% chance of being filled with 50% green balls.We pull 3 green balls in a row from the urn. The chances of that happening with an urncomposed of 50% green balls are (0.5*0.5*0.5) = 0.125. When we plug the numbers intothe formula we get pr(0.5|0.125) = (0.125*0.5)/0.125 = 0.5 or 50%, which I know isincorrect. I don’t know where I made the mistake though. The book comes up with 8%somehow. I don’t know where I am going wrong.
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