Write about why you would like to be this age and what you would do.

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?

What exactly would you like to do and why would you do it?

Why would you like to do this, and what would you like to do?

We read Julius Caesar that year (still one of my favorite plays of all time, by the way!), and even back than I found it to be a wonderful, character-driven drama; I mostly loved the character of Cassius, and I re-read his dialogue carefully, trying to understand his rhetorical strategies as he convinced Brutus to kill his friend--Caesar--for the good of the government. As we got deeper into the play, I wanted to write about Cassius and Brutus during those 10-20 minutes we were given for our journals, but I couldn't; instead, I was forced to write to our teacher's prompts, which sounded something like --"Do you believe in prophecy? Why or why not? If so, what convinced you? If not, what would change your mind?" See, my tenth grade teacher wanted us to focus in on the famous quotes from the play, like "Beware the Ides of March," which explains the type of journal prompts he was giving us. My teacher wanted us to write quietly, then he wanted to share all of his own personal stories about why he kind of believed in prophecy. I had no problem discussing his area of interest from the play--prophecy--, but years later I can't help but think that we could have had some much richer whole-class, socratic seminars--or heck, even just informal discussions--if we had a choice to a) respond to the teacher's prompt, or to b) explore a different literature-based idea that we could bring to the table based on what we were finding interesting in the literature. How hard would giving us a choice have been for him? What always struck me as the most interesting thing about that teacher's Julius Caesar unit was that everyone in my class was assigned the exact same essay topic as our summative assessment to the unit; it was something like, "How do the dreams of men and the idea of prophecy shape our thinking about the future?" I wrote a lackluster essay, I'm sure, because I didn't care about that topic; now, had he allowed me to write about Cassius and his persuasive skills, I would have given him a killer essay. I truly would have.

What would you like to invent? | 2bitsworthofthoughts

Note, however, that "No problem should ever have to be solvedtwice." does not imply that you have to consider all existingsolutions sacred, or that there is only one right solution to anygiven problem. Often, we learn a lot about the problem that we didn'tknow before by studying the first cut at a solution. It's OK, andoften necessary, to decide that we can do better. What's not OK isartificial technical, legal, or institutional barriers (likeclosed-source code) that prevent a good solution from being re-usedand people to re-invent wheels.