## Probability Theory | Math Goodies

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:

^{4}^{4}which 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?

## 254A, Notes 0: A review of probability theory | What's …

Last updated: 2006.06.04 Yudkowsky's "Intuitive Explanation of Bayesian Reasoning" and Rovner's"BayesApplet" may both be freely used by any nonprofit organization oreducational institution. No royalties or per-page charges arenecessary to reproduce this document as course materials, either inprinted form or online.

## Probability Sampling - Social Research Methods

Jaynes, in "Probability Theory With Applications in Science andEngineering", suggests that credibility and evidence should be measuredin decibels.

Decibels?

Decibels are used for measuring exponential differences ofintensity. For example, if the sound from an automobile horncarries 10,000 times as much energy (per square meter per second) asthe sound from an alarm clock, the automobile horn would be 40 decibelslouder. The sound of a bird singing might carry 1,000 times lessenergy than an alarm clock, and hence would be 30 decibelssofter. To get the number of decibels,you take the logarithm base 10 and multiply by 10.

Suppose we start with a prior probability of 1% that a woman has breastcancer, corresponding to an odds ratio of 1:99. And then weadminister three tests of likelihood ratios 25:3, 18:1, and 7:2. You multiply thosenumbers...