Related: Somebody flips a coin 100 times. It’s come up heads each time. What are the odds it comes up heads on the 101st throw? If you’re doing a probability problem, the answer is 50%. But in reality, would you bet even with 1000x1 odds in your favor on that 101st throw being a tails?
Well no, the answer isn’t 50%. Apply Bayes Theorem, using 0.5 as the prior and the 100 coinflips as the conditional probability, and you basically get 1-epsilon, because the coin is most likely biased
This depends on your prior for the occurrence of biased coins! If you have 10 coins of which one is two-headed and the others normal, and you draw one and start flipping, it doesn’t take many flips to be pretty sure you have the two-headed coin. But if biased coins are very rare, it takes a lot more flips.
Given 2^100 odds, it’s more likely the person flipping the coin is using the double-headed quarter I tossed into a wishing well in Mexico ten years previously than that the flips were entirely natural.
It works out to 6.338*10^29 against, assuming we’re not favoring a series of heads over tails. At those odds, a casting mistake resulting in a chunk of ferrous material being embedded in the coin and a magnetic anomaly caused by the alignment of the microwave and the toaster and the fact that the television happens to be tuned to channel 29 with a volume setting of 9 start to become viable contenders as explanations.
I meant “The ‘correct’ answer when you’re taking a probability class or are taking part in a study examining gambling fallacies and don’t want to be counted as one of the people who ‘clearly’ doesn’t understand probabilities.”
That said, I actually had a decent probability and statistics coursework instructor who would have marked “~100%” correct if a decent explanation were provided for the answer. (I answered problems that way all the time, although I don’t think that exact question turned up in any tests even though it did turn up in the probability textbook.)
I never took any test with that question with “a coin 100 times”, but I did have one with red/black on a roulette wheel (which I assume would be much harder to fudge than a coin) ten (IIRC) times. (And I answered 18⁄37 -- the zero is neither black nor red).
You forgot the 00, although it depends on whether you’re playing European or US roulette.
Actually, I love roulette. And yes, it’s much harder to fudge than a coin; the best that can be managed would be better described as a “nudge”—the timing of the dealer’s throw can make a tiny % of difference.
The skill necessary to land the ball exactly where the dealer wants it would be superhuman, but as my dad commented (describing practicing with throwing knives and throwing stars), practice has a tendency to make you luckier. (Yes, my homeschooling lessons involved throwing knives.)
Well no, the answer isn’t 50%. Apply Bayes Theorem, using 0.5 as the prior and the 100 coinflips as the conditional probability, and you basically get 1-epsilon, because the coin is most likely biased
This depends on your prior for the occurrence of biased coins! If you have 10 coins of which one is two-headed and the others normal, and you draw one and start flipping, it doesn’t take many flips to be pretty sure you have the two-headed coin. But if biased coins are very rare, it takes a lot more flips.
Given 2^100 odds, it’s more likely the person flipping the coin is using the double-headed quarter I tossed into a wishing well in Mexico ten years previously than that the flips were entirely natural.
It works out to 6.338*10^29 against, assuming we’re not favoring a series of heads over tails. At those odds, a casting mistake resulting in a chunk of ferrous material being embedded in the coin and a magnetic anomaly caused by the alignment of the microwave and the toaster and the fact that the television happens to be tuned to channel 29 with a volume setting of 9 start to become viable contenders as explanations.
If I understand correctly what he meant by “a probability problem”, your prior that the coin is biased is 0.
I meant “The ‘correct’ answer when you’re taking a probability class or are taking part in a study examining gambling fallacies and don’t want to be counted as one of the people who ‘clearly’ doesn’t understand probabilities.”
That said, I actually had a decent probability and statistics coursework instructor who would have marked “~100%” correct if a decent explanation were provided for the answer. (I answered problems that way all the time, although I don’t think that exact question turned up in any tests even though it did turn up in the probability textbook.)
I never took any test with that question with “a coin 100 times”, but I did have one with red/black on a roulette wheel (which I assume would be much harder to fudge than a coin) ten (IIRC) times. (And I answered 18⁄37 -- the zero is neither black nor red).
You forgot the 00, although it depends on whether you’re playing European or US roulette.
Actually, I love roulette. And yes, it’s much harder to fudge than a coin; the best that can be managed would be better described as a “nudge”—the timing of the dealer’s throw can make a tiny % of difference.
The skill necessary to land the ball exactly where the dealer wants it would be superhuman, but as my dad commented (describing practicing with throwing knives and throwing stars), practice has a tendency to make you luckier. (Yes, my homeschooling lessons involved throwing knives.)