// ODDS = YEP:NOPE
YEP, NOPE = MAKE UP SOME INITIAL ODDS WHO CARES
FOR EACH E IN EVIDENCE
YEP *= CHANCE OF E IF YEP
NOPE *= CHANCE OF E IF NOPE
ASSUME E // DO NOT DOUBLE COUNT
The thing to remember is that yeps and nopes never cross. The colon is a thick & rubbery barrier. Yep with yep and nope with nope.
bear : notbear = 1:100 odds to encounter a bear on a camping trip around here in general * 20% a bear would then scratch my tent : 50% a notbear would * 10% a bear would then flip my tent over : 1% a notbear would * 95% a bear would then look exactly like a fucking bear inside my tent : 1% a notbear would * 0.01% chance a bear would then eat me alive : 0.001% chance a notbear would
As you die you conclude 1*20*10*95*.01 : 100*50*1*1*.001 = 190 : 5 odds that a bear is eating you.
Here's a visual description: Imagine all worlds, before you see evidence cut into two: YEP and NOPE. The ratio of how many are in each (aka probability mass or size) represents the prior odds. Now, you see some evidence E (e.g. a metal detector beeping), so we want to know the ratio after seeing it.
Each part of the prior cut produces worlds with E (e.g. produces beeps). A YEP produces (Chance of E if YEP) amount of E worlds while a NOPE produces (Chance of E with NOPE).
And thus the new ratio is the product.
In case you don't know what odds are, they express a ratio using a pair of numbers where the overall scale is irrelevant, e.g. 1:2 and 2:4 represent the same ratio. Probabilities are the values when you scale so that the sum over all outcomes is 1, so in this case 1:2 = 1/3 : 2/3 so the probabilities are 1/3, 2/3.
In my opinion, the odds form is the superior form, because it's very easy to use and remember and "philosophically speaking" relative probabilityness is possibly more fundamental. Even at higher levels it's often more practical. I see it as a pedagogical mistake that Bayes theorem is usually first explained in probability form - even on this site! Basic things should be deeply understood by ~everyone.
The thing to remember is that yeps and nopes never cross. The colon is a thick & rubbery barrier. Yep with yep and nope with nope.
bear : notbear =
1:100 odds to encounter a bear on a camping trip around here in general
* 20% a bear would then scratch my tent : 50% a notbear would
* 10% a bear would then flip my tent over : 1% a notbear would
* 95% a bear would then look exactly like a fucking bear inside my tent : 1% a notbear would
* 0.01% chance a bear would then eat me alive : 0.001% chance a notbear would
As you die you conclude 1*20*10*95*.01 : 100*50*1*1*.001 = 190 : 5 odds that a bear is eating you.