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 1⁄6, 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.
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.
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 1⁄6, 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.