1: I submit that I merely tested one hypothesis—that the keys were in my house—and sampled with replacement. While non-optimal, the use of a probabilistic search algorithm has precedent, e.g. in paleontology and astronomy.
(a) No less confidence than one would have in the theory of evolution. The manner of finding support among the fossil record consists of excavating likely locations, and in a like manner I have performed a search of the likely location for my keys, being my domicile, and proceeded until all locations were exhausted.
(b) The uncertainty of my keys being in my pocket is no greater than the uncertainty in the door being locked (as my door only has a sliding bolt that requires a key to operate). If the probability of my keys not being in my pocket is a (being less than or equal to the probability of the door failing to lock), then the probability of failing to open the door is a(1-a) = a—a^2. This reaches its maximum value at a = 1⁄2, which still leaves me with a 3⁄4 chance of entering my house.
2:
(a) Naturally. As my Internet use generally has a negative value, anything with a positive expectation can only be a benefit.
(b) We wish to compute the probability of winning given the event that I observed the winning numbers on the lottery website. Suppose, then, that a given observation of the winning lottery numbers has a 95% chance of being correct. This can be found by a trivial application of Bayes’ theorem, obtaining 19 in 1,000,000, which is roughly one part in 50 thousand. The question, then, becomes: how many independent observations would it take for the expected probability of winning make it worth submitting a claim to the lottery office? I leave this as an exercise.
Alternative hypotheses include: misreading the page, looking at the wrong drawings, erroneous information on the page, and pranks by some rogue.
(c) Once I’m able to spend the money, I expect to be far too intoxicated to ponder the metaphysics of the matter.
1: I submit that I merely tested one hypothesis—that the keys were in my house—and sampled with replacement. While non-optimal, the use of a probabilistic search algorithm has precedent, e.g. in paleontology and astronomy.
(a) No less confidence than one would have in the theory of evolution. The manner of finding support among the fossil record consists of excavating likely locations, and in a like manner I have performed a search of the likely location for my keys, being my domicile, and proceeded until all locations were exhausted.
(b) The uncertainty of my keys being in my pocket is no greater than the uncertainty in the door being locked (as my door only has a sliding bolt that requires a key to operate). If the probability of my keys not being in my pocket is a (being less than or equal to the probability of the door failing to lock), then the probability of failing to open the door is a(1-a) = a—a^2. This reaches its maximum value at a = 1⁄2, which still leaves me with a 3⁄4 chance of entering my house.
2: (a) Naturally. As my Internet use generally has a negative value, anything with a positive expectation can only be a benefit.
(b) We wish to compute the probability of winning given the event that I observed the winning numbers on the lottery website. Suppose, then, that a given observation of the winning lottery numbers has a 95% chance of being correct. This can be found by a trivial application of Bayes’ theorem, obtaining 19 in 1,000,000, which is roughly one part in 50 thousand. The question, then, becomes: how many independent observations would it take for the expected probability of winning make it worth submitting a claim to the lottery office? I leave this as an exercise.
Alternative hypotheses include: misreading the page, looking at the wrong drawings, erroneous information on the page, and pranks by some rogue.
(c) Once I’m able to spend the money, I expect to be far too intoxicated to ponder the metaphysics of the matter.