What is the point of this thing? If you observe new evidence

it's like this: you have a map in front of you where you want to locate an object, you can guess where the object is with some probability. So you draw a large circle somewhere from your calculations. Then new info comes up and your new map is smaller, so you redo your calculations and get a smaller "more precise" circle. Basically the new info you got helped you in locating your object by shrinking your search area.

>>8477296
It can give you an understanding of how unintuitive probabilities are at times.

e.g.
If a city has 1000 taxicabs, and 100 of these are green, the rest are yellow.
A person is hit by what is reported as a green taxi.

When we check this person in similar light-conditions we find they correctly identify the color as green 90% of the time.

Let's enter these into the equation:
P(B) - probability of correct identification (=0.9)
P(A) - probability of being a green cab. (= 0.1)
P(B|A) - probability of identifying correct given green cab.
P(A|B) - probability of green cab given identified as green.

So; What's the probability the cab that ran over the person was really green ?

>>8477328
Ps. I have here also assumed they were 90% correct for the identification of yellow cabs too.

Which gives us the equation:
P(A|B) = (0.9 * 0.1) /0.9 = 0.1, or 10%

For a twist; let's say yellow cabs are correctly identified 99% of the time, and green cabs are identified correctly 90% of the time.

This changes the equations a bit.
P(B) = identified as green.
P(not B) - identified as not green.
P(A) - Green (0.1)
P(not a) . not green.(0.9)
P(B|A) - identified as green, is green. ( 0.9)
P(B| not A) - identified as green, not green. (0.01)
P(not B | A) - identified as yellow,, is green (0.1)
P(not B | not A) - identified as yellow, is yellow (0.99)

P(A | B) - is green, identified as green (P = ?)

>>8477328
Wow, that is unintuitive, but it does make sense.

>>8477296
its conditional probability

its the chance of something happening given that you know something that will affect it

so think about things like the montee hall problem, which is a conditional probability problem

its the same thing

if you KNOW some new information, it changes your odds

if you dont know, it doesnt matter what door you pick

basically its unintuitive because people don't consider that it's changing the initial odds

"why would changing your door help?" is because they arent realising that the problem has now changed and you have to consider a new problem.