Lightswitches
There is probably some obvious solution to this puzzle, but it eludes me. I’m not sure how to plug it into the equation for Bayes’ Theorem. And the situation described happened last August, so I’m probably not going to figure it out on my own.
There are two lightswitches next to each other, and they control two lights (which have no other switches connected to them). I have used the switches a few times before, but don’t occurrently recall which switch goes to which light, or whether the up or down position is the one that signifies off-ness. One light is on, one light is off, and the switches are in different positions. I want both lights off. So I guess a switch, and I’m right. What should be my credence be that my previous experience with this set of lightswitches helped me guess correctly, given that I felt like I was guessing at random (and would have had a 50% shot at being right were that the case)? How much would this be different if I’d guessed wrong the first time?
P(experience helped | guessed right) = P(experience helped) * P(guessed right | experience helped) / P(guessed right)
Bayes’ theorem updates on evidence by multiplying the old probability by the new evidence. The new evidence comes in the form of how likely the observation was under your hypothesis compared to the general rate.
So what comes out of bayes’ theorem here isn’t exactly an answer, but is an answer for each possible degree of “experience helped,” where you can then take the average to find out the likeliest helpfulness of experience.
P(guessed right) = 0.5. So if, for example, experience boosts you up to guessing right with P=0.6, then your prior probability rises by 0.6 / 0.5, while if you guessed wrong it would drop your prior probability by a factor of (1-0.6) / 0.5.
If you approximate experience as helping your probability of guessing right up to some value that’s uniformly distributed between 0.5 and 1, we (pretty obviously) get an average of 0.75. So guessing right multiplies P(experience helped) by a factor of approximately 3⁄2, while guessing wrong multiplies it by a factor of approximately 1⁄2.
What should your prior value of P(experience helped) be? That’s more complicated, since it deals with your memories and associated junk. It probably shouldn’t be more than 1⁄2, to be compatible with the guess I made a paragraph ago that P(guessed right | experience helped) can get close to 1 (that matters—check Bayes’ theorem), so if we do the same uniform approximation between 0 and 1⁄2 we get something like 0.25, going up to 0.375 if you guess right or down to 0.125 if you guess wrong.
EDIT: not quite right, see Misha’s comment.
P(guessed right) isn’t exactly 0.5 (most likely). We expand it as
P(experience helped (prior)) P(guessed right | experience helped) + P(experience didn’t help (prior)) P(guessed right | experience didn’t help).
Of these, only P(guessed right | experience didn’t help) should be 0.5; P(guessed right | experience helped) should be higher. So on average P(guessed right) is somewhere in between depending on your priors.
Right, thanks. Pushing the uniform approximations to the point of silliness I get 4⁄7 for a rough value (with the prior then being 2⁄7).
This was the paragraph I needed, thanks.
If I encountered this situation with no prior experience with the particular switches and lights, I expect I could reliably turn the light off by flipping the switch in the up position to the down position. Most switches are oriented that way.
Other switches in the same dwelling are oriented either way with no obvious bias towards either.
The presence of oddly oriented light switches is a red flag for other non-professional electrical work...
Are many of them 3-way switches (2 switches that control the same light)?
For those “up” and “down” aren’t anything special; for them to work the on-state of the light depends on the state of both switches (such that toggling either toggles the state of the light).
Most of the ones in my house are 3-way (though for most of the lights we only use one of the relevant pair). It’ll be interesting to see if it takes my kids longer than usual to learn the usual “up=on, down=off” rule.
Not sure. I don’t live there, I was just crashing in the apartment for a little under a week and didn’t have cause to learn all the lightswitches.