You put your MP3 player on random. You have a playlist of 20 songs. What are the odds that the next song played is the same song which was just played?
In most people’s experience, random sequences aren’t. It’s the idea of true randomness which doesn’t conform to the reality people live in.
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?
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%.
If the coin is fair, the answer is 50%… but what are the odds that the coin is fair, given that it’s come up heads 100 times?
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.)
You put your MP3 player on random. You have a playlist of 20 songs. What are the odds that the next song played is the same song which was just played?
I think the option is more typically called “shuffle”, which actually accurately represents what it does.
I once had a MP3 player where “random” actually did what it said (for some value of “actually”), including playing the same song twice in a row once in a while.
I did too (up to and including the “some value of actually,” as it played 20% of the songs 90% of the time), which is why I brought up that example. It annoyed me to no end that the button did what it said it did.
In general, interfaces should reflect what the user expects the interface to be, rather than what the designer expects the user to interpret the interface to be.
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?
In reality, I wouldn’t bet either way. It’s very likely that there’s a trick where the coinflip will go against whichever way I bet.
You put your MP3 player on random. You have a playlist of 20 songs. What are the odds that the next song played is the same song which was just played?
In most people’s experience, random sequences aren’t. It’s the idea of true randomness which doesn’t conform to the reality people live in.
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?
If the coin is fair, the answer is 50%… but what are the odds that the coin is fair, given that it’s come up heads 100 times?
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.)
I think the option is more typically called “shuffle”, which actually accurately represents what it does.
I once had a MP3 player where “random” actually did what it said (for some value of “actually”), including playing the same song twice in a row once in a while.
I did too (up to and including the “some value of actually,” as it played 20% of the songs 90% of the time), which is why I brought up that example. It annoyed me to no end that the button did what it said it did.
In general, interfaces should reflect what the user expects the interface to be, rather than what the designer expects the user to interpret the interface to be.
Actually, I kind-of liked its unpredictability. When I didn’t want to listen to a song for a second time (and sometimes I did), I just skipped it.
In reality, I wouldn’t bet either way. It’s very likely that there’s a trick where the coinflip will go against whichever way I bet.
So, you no box on Newcomb’s Problem? :)