• While the prob­a­bil­is­tic rea­son­ing em­ployed in the card ques­tion is cor­rect and fits in with your over­all point, it’s rather la­bor-in­ten­sive to ac­tu­ally think through.

In or­der to get two red cards, you need to pick the right pair of cards. Only one pair will do. There are six ways to pick a pair of cards out of a group of 4 (when, as here, or­der doesn’t mat­ter). There­fore, the odds are 16, as one out of the six pos­si­ble pairs you’ll pick will be the cor­rect pair.

Similarly, we know the weath­er­per­son cor­rectly pre­dicts 12.5% of days that will be rainy. We know that 20% of days will ac­tu­ally be rain­ing. That gives us “12.5/​20 = 5/​8” pretty quickly. Grind­ing our way through all the P(X [ ~X) rep­re­sen­ta­tion makes a sim­ple and in­tu­itive calcu­la­tion look re­ally in­timi­dat­ing.

I’m not en­tirely sure of your pur­pose in this se­quence, but it seems to be to im­prove peo­ple’s prob­a­bil­is­tic rea­son­ing. Ex­plain­ing prob­a­bil­ities through this long and de­tailed method seems guaran­teed to fail. Peo­ple who are perfectly com­fortable with such com­plex ex­pla­na­tions gen­er­ally already get their ap­pli­ca­tion. Peo­ple who are not so com­fortable throw up their hands and stick with their gut. I sus­pect that a large part of the ex­pla­na­tion of math­e­mat­i­cal illiter­acy is that peo­ple aren’t ac­tu­ally taught how to ap­ply math­e­mat­ics in any prac­ti­cal sense; they’re given a log­i­cally rigor­ous and for­mal proof in un­nec­es­sary de­tail which is too com­plex to use in in­for­mal rea­son­ing.

• Speak­ing only for my­self, I’m in that awk­ward mid­dle stage—I un­der­stand prob­a­bil­ity well enough to solve toy prob­lems, and to fol­low ex­pla­na­tions of it in real prob­lems, but not enough to be con­fi­dent in my own prob­a­bil­is­tic in­ter­pre­ta­tion of new prob­lem do­mains. I’m look­ing for­ward to this se­quence as part of my ed­u­ca­tion and definitely ap­pre­ci­ate see­ing the for­mal­ity be­hind the ap­pli­ca­tions.

• I’m glad this is in­tu­itive for you!

The rea­son I spotlighted la­bor-in­ten­sive meth­ods is be­cause this post is tar­geted at peo­ple who don’t find this in­tu­itive. I’d rather give them a method that can be ex­tended to other situ­a­tions with low risk (ap­ply­ing Bayes’ Rule, imag­in­ing the world af­ter re­ceiv­ing an up­date and calcu­lat­ing new prob­a­bil­ities) rather than iden­ti­fy­ing sym­me­tries in the prob­lems and us­ing those to quickly get an­swers.

The rest of the se­quence uses this as back­ground, but prob­a­bil­ity calcu­la­tions play a sec­ondary role. The tech­niques I’ll dis­cuss re­quire a mod­er­ate level of com­fort with prob­a­bil­ities, but not with prob­a­bil­is­tic calcu­la­tions- those can (and prob­a­bly should) be offloaded to a calcu­la­tor. The challenge is set­ting up the right prob­lem, not solv­ing a prob­lem once you’ve set it up.