Mixing
Probability of probability / Confidence in probability
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Actually I got a few questions on the same topic... I think they are related in the sense that, how do we infer information about an unknown distribution with limited samples.
-
There's a coin that no one knows whether it's fair on both sides, i.e. it can be exactly fair or it can be biased to any extent. A person flipped the coin three times and all three times it landed on the head.
a. What is the chance that this coin is biased such that it's more likely to land on head than on tail?
b. What is the most probable extent of its biasness (is it more probable that this coin is [90% head 10% tail] than [80% head 20% tail]), or is it impossible to tell with the available information?
-
There are R balls in a box (R>9 and finite). N or them is/are white and R-N of them is/are black. A person randomly drew out 10 of the balls one by one (so there's one less ball every time he drew) and it turns out all of them are white.
a. What is the chance that there were more white balls in the box than black balls?
b. What is the most probable ratio between the number of white balls and black balls?
-
A scientist discovered a new unstable isotope, and possesses 10 atoms of it. The scientist observed that after each exact minutes, one of the atoms decayed, such that after exactly ten minutes, all of them decayed.
a. What is the most probable decay constant of this isotope, and to what confidence level can the scientist say that this is the decay constant of this isotope?
b. To what confidence level can the scientist infers that this isotope follows an exponential decay (instead of a linear one)?
probability statistics confidence-interval
asked Jan 19 at 16:14
JackyJacky
205
$endgroup$
add a comment |
$begingroup$
Actually I got a few questions on the same topic... I think they are related in the sense that, how do we infer information about an unknown distribution with limited samples.
-
There's a coin that no one knows whether it's fair on both sides, i.e. it can be exactly fair or it can be biased to any extent. A person flipped the coin three times and all three times it landed on the head.
a. What is the chance that this coin is biased such that it's more likely to land on head than on tail?
b. What is the most probable extent of its biasness (is it more probable that this coin is [90% head 10% tail] than [80% head 20% tail]), or is it impossible to tell with the available information?
-
There are R balls in a box (R>9 and finite). N or them is/are white and R-N of them is/are black. A person randomly drew out 10 of the balls one by one (so there's one less ball every time he drew) and it turns out all of them are white.
a. What is the chance that there were more white balls in the box than black balls?
b. What is the most probable ratio between the number of white balls and black balls?
-
A scientist discovered a new unstable isotope, and possesses 10 atoms of it. The scientist observed that after each exact minutes, one of the atoms decayed, such that after exactly ten minutes, all of them decayed.
a. What is the most probable decay constant of this isotope, and to what confidence level can the scientist say that this is the decay constant of this isotope?
b. To what confidence level can the scientist infers that this isotope follows an exponential decay (instead of a linear one)?
probability statistics confidence-interval
asked Jan 19 at 16:14
JackyJacky
205
$endgroup$
$begingroup$
This Wikipedia article on Maximum Likelihood Estimation might be of interest: en.m.wikipedia.org/wiki/Maximum_likelihood_estimation
$endgroup$
– Simon
Jan 19 at 16:42
$begingroup$
Some answers will differ depending on whether your framework is 'frequentist' or 'Bayesian' probability. Can you give some context? // I am not a physicist, but I know enough to question whether the description of decay in #3 obeys any known rules. A more likely model is something like each molecule decays according to a Poisson process at the avg rate of 0.1 per min.
$endgroup$
– BruceET
Jan 22 at 1:47
add a comment |
$begingroup$
Actually I got a few questions on the same topic... I think they are related in the sense that, how do we infer information about an unknown distribution with limited samples.
-
There's a coin that no one knows whether it's fair on both sides, i.e. it can be exactly fair or it can be biased to any extent. A person flipped the coin three times and all three times it landed on the head.
a. What is the chance that this coin is biased such that it's more likely to land on head than on tail?
b. What is the most probable extent of its biasness (is it more probable that this coin is [90% head 10% tail] than [80% head 20% tail]), or is it impossible to tell with the available information?
-
There are R balls in a box (R>9 and finite). N or them is/are white and R-N of them is/are black. A person randomly drew out 10 of the balls one by one (so there's one less ball every time he drew) and it turns out all of them are white.
a. What is the chance that there were more white balls in the box than black balls?
b. What is the most probable ratio between the number of white balls and black balls?
-
A scientist discovered a new unstable isotope, and possesses 10 atoms of it. The scientist observed that after each exact minutes, one of the atoms decayed, such that after exactly ten minutes, all of them decayed.
a. What is the most probable decay constant of this isotope, and to what confidence level can the scientist say that this is the decay constant of this isotope?
b. To what confidence level can the scientist infers that this isotope follows an exponential decay (instead of a linear one)?
probability statistics confidence-interval
asked Jan 19 at 16:14
JackyJacky
205
$endgroup$
Actually I got a few questions on the same topic... I think they are related in the sense that, how do we infer information about an unknown distribution with limited samples.
-
There's a coin that no one knows whether it's fair on both sides, i.e. it can be exactly fair or it can be biased to any extent. A person flipped the coin three times and all three times it landed on the head.
a. What is the chance that this coin is biased such that it's more likely to land on head than on tail?
b. What is the most probable extent of its biasness (is it more probable that this coin is [90% head 10% tail] than [80% head 20% tail]), or is it impossible to tell with the available information?
-
There are R balls in a box (R>9 and finite). N or them is/are white and R-N of them is/are black. A person randomly drew out 10 of the balls one by one (so there's one less ball every time he drew) and it turns out all of them are white.
a. What is the chance that there were more white balls in the box than black balls?
b. What is the most probable ratio between the number of white balls and black balls?
-
A scientist discovered a new unstable isotope, and possesses 10 atoms of it. The scientist observed that after each exact minutes, one of the atoms decayed, such that after exactly ten minutes, all of them decayed.
a. What is the most probable decay constant of this isotope, and to what confidence level can the scientist say that this is the decay constant of this isotope?
b. To what confidence level can the scientist infers that this isotope follows an exponential decay (instead of a linear one)?
probability statistics confidence-interval
probability statistics confidence-interval
asked Jan 19 at 16:14
JackyJacky
205
asked Jan 19 at 16:14
JackyJacky
205
edited Jan 19 at 16:32
Jacky
asked Jan 19 at 16:14
JackyJacky
205
asked Jan 19 at 16:14
JackyJacky
205
asked Jan 19 at 16:14
JackyJacky
205
205
$begingroup$
This Wikipedia article on Maximum Likelihood Estimation might be of interest: en.m.wikipedia.org/wiki/Maximum_likelihood_estimation
$endgroup$
– Simon
Jan 19 at 16:42
$begingroup$
Some answers will differ depending on whether your framework is 'frequentist' or 'Bayesian' probability. Can you give some context? // I am not a physicist, but I know enough to question whether the description of decay in #3 obeys any known rules. A more likely model is something like each molecule decays according to a Poisson process at the avg rate of 0.1 per min.
$endgroup$
– BruceET
Jan 22 at 1:47
add a comment |
$begingroup$
This Wikipedia article on Maximum Likelihood Estimation might be of interest: en.m.wikipedia.org/wiki/Maximum_likelihood_estimation
$endgroup$
– Simon
Jan 19 at 16:42
$begingroup$
Some answers will differ depending on whether your framework is 'frequentist' or 'Bayesian' probability. Can you give some context? // I am not a physicist, but I know enough to question whether the description of decay in #3 obeys any known rules. A more likely model is something like each molecule decays according to a Poisson process at the avg rate of 0.1 per min.
$endgroup$
– BruceET
Jan 22 at 1:47
$begingroup$
This Wikipedia article on Maximum Likelihood Estimation might be of interest: en.m.wikipedia.org/wiki/Maximum_likelihood_estimation
$endgroup$
– Simon
Jan 19 at 16:42
$begingroup$
This Wikipedia article on Maximum Likelihood Estimation might be of interest: en.m.wikipedia.org/wiki/Maximum_likelihood_estimation
$endgroup$
– Simon
Jan 19 at 16:42
$begingroup$
Some answers will differ depending on whether your framework is 'frequentist' or 'Bayesian' probability. Can you give some context? // I am not a physicist, but I know enough to question whether the description of decay in #3 obeys any known rules. A more likely model is something like each molecule decays according to a Poisson process at the avg rate of 0.1 per min.
$endgroup$
– BruceET
Jan 22 at 1:47
$begingroup$
Some answers will differ depending on whether your framework is 'frequentist' or 'Bayesian' probability. Can you give some context? // I am not a physicist, but I know enough to question whether the description of decay in #3 obeys any known rules. A more likely model is something like each molecule decays according to a Poisson process at the avg rate of 0.1 per min.
$endgroup$
– BruceET
Jan 22 at 1:47
add a comment |
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$begingroup$
This Wikipedia article on Maximum Likelihood Estimation might be of interest: en.m.wikipedia.org/wiki/Maximum_likelihood_estimation
$endgroup$
– Simon
Jan 19 at 16:42
$begingroup$
Some answers will differ depending on whether your framework is 'frequentist' or 'Bayesian' probability. Can you give some context? // I am not a physicist, but I know enough to question whether the description of decay in #3 obeys any known rules. A more likely model is something like each molecule decays according to a Poisson process at the avg rate of 0.1 per min.
$endgroup$
– BruceET
Jan 22 at 1:47