"TheBestNotes on A Long Way Gone"
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In a reverse way, the tragedy of the commons reappears inproblems of pollution. Here it is not a question of takingsomething out of the commons, but of putting something in --sewage, or chemical, radioactive, and heat wastes into water;noxious and dangerous fumes into the air; and distracting andunpleasant advertising signs into the line of sight. Thecalculations of utility are much the same as before. The rationalman finds that his share of the cost of the wastes he dischargesinto the commons is less than the cost of purifying his wastesbefore releasing them. Since this is true for everyone, we arelocked into a system of "fouling our own nest," so longas we behave only as independent, rational, free enterprisers.
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or you could just add their logarithms:
It starts out as fairly unlikely that a woman has breast cancer - ourcredibility level is at -20 decibels. Then three test resultscomein, corresponding to 9, 13, and 5 decibels of evidence. Thisraises the credibility level by a total of 27 decibels, meaning thattheprior credibility of -20 decibels goes to a posterior credibility of 7decibels. So the odds go from 1:99 to 5:1, and the probabilitygoes from 1% to around 83%.
Just for fun, try and work this one out in your head. You don'tneed to be exact - a rough estimate is good enough. When you'reready, continue onward.
According to a study performed by Lawrence Phillips and Ward Edwards in1966, most people, faced with this problem, give an answer in the range70% to 80%. Did you give a substantially higher probability thanthat? If you did, congratulations - Ward Edwards wrote that veryseldom does a person answer this question properly, even if the personis relatively familiar with Bayesian reasoning. The correctansweris 97%.
The likelihood ratio for the test result "red chip" is 7/3, while thelikelihood ratio for the test result "blue chip" is 3/7. Thereforea blue chip is exactly the same amount of evidence as a red chip, justin the other direction - a red chip is 3.6 decibels of evidence for thered bag, and a blue chip is -3.6 decibels of evidence. If youdrawone blue chip and one red chip, they cancel out. So the of red chips to blue chipsdoes not matter; only the of red chips over blue chips matters. There were eight red chipsand four blue chips in twelve samples; therefore, four red chips than bluechips. Thus the posterior odds will be:
44which is around 30:1, i.e., around 97%.
The prior credibility starts at 0 decibels and there's a total ofaround 14 decibels of evidence, and indeed this corresponds to odds ofaround 25:1 or around 96%. Again, there's some rounding error,butif you performed the operations using exact arithmetic, the resultswould be identical.
We can now see that the bookbag problem would have exactly the same answer, obtainedinjust the same way, if sixteen chips were sampled and we found ten redchips and six blue chips.
What is the sequence of arithmetical operations that you performed tosolve this problem?
(45%*30%) / (45%*30% + 5%*70%)
Similarly, to find the chance that a woman with positive mammographyhas breast cancer, we computed:
The fully general form of this calculation is known as or
Given some phenomenon A that we want to investigate, and an observationX that is evidence about A - for example, in the previous example, A isbreast cancer and X is a positive mammography - Bayes' Theorem tells ushow we should ourprobability of A, given the X.
By this point, Bayes' Theorem may seem blatantly obvious or eventautological, rather than exciting and new. If so, thisintroduction has in its purpose.
So why is it that some people are so about Bayes' Theorem?
"Do you believe that a nuclear war will occur in the next 20 years?