a thoughtful web.
comment
wasoxygen  ·  754 days ago  ·  link  ·    ·  parent  ·  post: Probably Overthinking It: Bayes' Theorem is not optional

one out of nine chance of rain being in Seattle

They say it rains about 10% of the time, and "A base rate of 10% corresponds to prior odds of 1:9." I found this notation a bit confusing, but I think it corresponds to a one-in-ten chance of rain. For each rainy day, you get nine sunny days.

The difference between probabilities expressed as a value between 0 and 1 and these "odds" if that's what 1:9 is called is likely part of my confusion in following the article.

It was actually Professor Brian who demonstrated how useful a simulation can be, while analyzing this bizarre problem:

Say you know a family has two children, and further that at least one of them is a girl named Florida. What is the probability that they have two girls?

But a simulator, based on a random number generator, seems like a good way to check our work. I still trust my numbers as long as I feel like I know what I am doing. Say we start with a convenient number of days:

``  270 days  243 sunny``   27 rainy``

Each friend will be expected to give the same ratio of answers. We call Albert first and he responds:

``  True  "no"  on 2/3 of 243 sunny days = 162 days  False "yes" on 1/3 of 243 sunny days =  81 days  True  "yes" on 2/3 of  27 rainy days =  18 days``  False "no"  on 1/3 of  27 rainy days =   9 days``

So Albert says "yes" on 99 days, and on 81 days it is sunny and on 18 days it is raining. We increase our expectation of rain from 10% to 18/99, about 18%.

We expect the same ratio of responses from Betty:

``  True  "no"  on 2/3 of 81 sunny days = 54 days  False "yes" on 1/3 of 81 sunny days = 27 days  True  "yes" on 2/3 of 18 rainy days = 12 days``  False "no"  on 1/3 of 18 rainy days =  6 days``

So Betty says "yes" on 39 days, and on 27 days it is sunny and on 12 days it is raining. We increase our expectation of rain from 18/99 to 12/39, about 31%.

We expect the same ratio of responses from Charlie:

``  True  "no"  on 2/3 of 27 sunny days = 18 days  False "yes" on 1/3 of 27 sunny days =  9 days  True  "yes" on 2/3 of 12 rainy days =  8 days``  False "no"  on 1/3 of 12 rainy days =  4 days``

So Charlie says "yes" on 17 days, and on 9 days it is sunny and on 8 days it is raining. We increase our expectation of rain from 12/39 to 8/17, about 47%.

This seems very clear and agrees with the given answer. But I still haven't used Bayes' theorem.