Rephrase the problem without the inferential step from “it will rain” and “sprinkler will turn on” to “lawn will be wet”. Just use “lawn will be wet” in all those places. We can do this because we are using symbolic logic.
works because each part of the condition “Saturday, predicted rain” is involved in a different causal path to the result we are interested in. With the coins, neither being a penny specifically, or being minted in 2008, causes the coin flips to result in heads, so the analogous equation does not work.
That’s right—but the problem is not that “you can only decompose a conjunction that way when it’s on the left side of the |, not when it’s on the right.” The reasoning is syntactically correct. The two cases are syntactically identical when you do the syntactic substitution I suggested above.
The two cases are syntactically identical because you have not explicitly explained how you meet the conditions that allow the decomposition you used, conditions which are not in fact met in the case of the coins. Your “syntactic substitution” removes the information that could be used to show you meet the conditions.
(Just to be sure, you did intend the original “proof” as a “find the error” exercise, right?)
1) P(lawn wet| Saturday, predicted rain) = P(lawn watered or rain| Saturday, predicted rain)
2) = 1 - P(not lawn watered and not rain| Saturday, predicted rain)
3) = 1 - P(not lawn watered | Saturday, predicted rain) * (not rain | Saturday, predicted rain)
4) = 1 - P(not lawn watered | Saturday) * (not rain | predicted rain)
What is the analogy to step 1 with the coins?
Rephrase the problem without the inferential step from “it will rain” and “sprinkler will turn on” to “lawn will be wet”. Just use “lawn will be wet” in all those places. We can do this because we are using symbolic logic.
That is throwing away the information that allows you to make the inference. The equation:
works because each part of the condition “Saturday, predicted rain” is involved in a different causal path to the result we are interested in. With the coins, neither being a penny specifically, or being minted in 2008, causes the coin flips to result in heads, so the analogous equation does not work.
That’s right—but the problem is not that “you can only decompose a conjunction that way when it’s on the left side of the |, not when it’s on the right.” The reasoning is syntactically correct. The two cases are syntactically identical when you do the syntactic substitution I suggested above.
Or… hmm, I may be confused.
The two cases are syntactically identical because you have not explicitly explained how you meet the conditions that allow the decomposition you used, conditions which are not in fact met in the case of the coins. Your “syntactic substitution” removes the information that could be used to show you meet the conditions.
(Just to be sure, you did intend the original “proof” as a “find the error” exercise, right?)
Yes—but also to help me figure out how you ask whether 2 data sets are independent when they don’t intersect.