Mixing
Concluding that a coin is biased depending on number of flipping trials
0
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I am given the task to check if a coin is biased to land on heads. The bias must exceed a certain threshold i.e. $p > 0.5 + epsilon$ for some given $epsilon$. I would like to know how the number of flips affects the certainty of the conclusion that the coin is biased according to this definition.
Concretely, if I am allowed to flip the coin $n$ times, what is the probability of a false positive (coin is not biased but I claim that it is) and a false negative (coin is biased but I fail to spot it) as a function of $n$ and $epsilon$?
probability statistical-inference
asked Jan 28 at 11:14
user1936752user1936752
5841515
$endgroup$
add a comment |
0
$begingroup$
I am given the task to check if a coin is biased to land on heads. The bias must exceed a certain threshold i.e. $p > 0.5 + epsilon$ for some given $epsilon$. I would like to know how the number of flips affects the certainty of the conclusion that the coin is biased according to this definition.
Concretely, if I am allowed to flip the coin $n$ times, what is the probability of a false positive (coin is not biased but I claim that it is) and a false negative (coin is biased but I fail to spot it) as a function of $n$ and $epsilon$?
probability statistical-inference
asked Jan 28 at 11:14
user1936752user1936752
5841515
$endgroup$
1
$begingroup$
Please take a look at the Binomial Test.
$endgroup$
– denklo
Jan 28 at 11:26
$begingroup$
The question is wrong. It is no more answerable than "How long is a piece of string?". To make a sensible question, more information is needed. If we know the prior probability that the coin is biased, then we can calculate a posterior probability of it being so from the result of any given number of flips.
$endgroup$
– John Bentin
Jan 28 at 12:18
$begingroup$
@JohnBentin, I took a look at the binomial test. Isn't it correct to have a null hypothesis that the coin is biased such that $p>0.5 +epsilon$ and the alternative hypothesis that it is biased but with $pleq 0.5 +epsilon$? I wish to accept/reject the null hypothesis. Is this a well defined question?
$endgroup$
– user1936752
Jan 28 at 13:43
$begingroup$
The question didn't make clear that the set-up was to test the null hypothesis $p>0.5+epsilon$ versus the opposite. For any $pin[0,Bbb,, 1]$, you can calculate the probability of the outcome being as extreme as, or more extreme than, what is observed given that the probability of a head is $p$. But this looks like a pretty meaningless exercise. Even this useless calculation becomes impossible if we weaken the assumption from fixing P(head) to merely P(head) > $0.5+epsilon$ without specifying a distribution for P(head) in $[0.5+epsilon,Bbb, ,1]$.
$endgroup$
– John Bentin
Jan 28 at 19:23
add a comment |
0
0
0
$begingroup$
I am given the task to check if a coin is biased to land on heads. The bias must exceed a certain threshold i.e. $p > 0.5 + epsilon$ for some given $epsilon$. I would like to know how the number of flips affects the certainty of the conclusion that the coin is biased according to this definition.
Concretely, if I am allowed to flip the coin $n$ times, what is the probability of a false positive (coin is not biased but I claim that it is) and a false negative (coin is biased but I fail to spot it) as a function of $n$ and $epsilon$?
probability statistical-inference
asked Jan 28 at 11:14
user1936752user1936752
5841515
$endgroup$
I am given the task to check if a coin is biased to land on heads. The bias must exceed a certain threshold i.e. $p > 0.5 + epsilon$ for some given $epsilon$. I would like to know how the number of flips affects the certainty of the conclusion that the coin is biased according to this definition.
Concretely, if I am allowed to flip the coin $n$ times, what is the probability of a false positive (coin is not biased but I claim that it is) and a false negative (coin is biased but I fail to spot it) as a function of $n$ and $epsilon$?
probability statistical-inference
probability statistical-inference
asked Jan 28 at 11:14
user1936752user1936752
5841515
asked Jan 28 at 11:14
user1936752user1936752
5841515
asked Jan 28 at 11:14
user1936752user1936752
5841515
asked Jan 28 at 11:14
user1936752user1936752
5841515
asked Jan 28 at 11:14
user1936752user1936752
5841515
5841515
1
$begingroup$
Please take a look at the Binomial Test.
$endgroup$
– denklo
Jan 28 at 11:26
$begingroup$
The question is wrong. It is no more answerable than "How long is a piece of string?". To make a sensible question, more information is needed. If we know the prior probability that the coin is biased, then we can calculate a posterior probability of it being so from the result of any given number of flips.
$endgroup$
– John Bentin
Jan 28 at 12:18
$begingroup$
@JohnBentin, I took a look at the binomial test. Isn't it correct to have a null hypothesis that the coin is biased such that $p>0.5 +epsilon$ and the alternative hypothesis that it is biased but with $pleq 0.5 +epsilon$? I wish to accept/reject the null hypothesis. Is this a well defined question?
$endgroup$
– user1936752
Jan 28 at 13:43
$begingroup$
The question didn't make clear that the set-up was to test the null hypothesis $p>0.5+epsilon$ versus the opposite. For any $pin[0,Bbb,, 1]$, you can calculate the probability of the outcome being as extreme as, or more extreme than, what is observed given that the probability of a head is $p$. But this looks like a pretty meaningless exercise. Even this useless calculation becomes impossible if we weaken the assumption from fixing P(head) to merely P(head) > $0.5+epsilon$ without specifying a distribution for P(head) in $[0.5+epsilon,Bbb, ,1]$.
$endgroup$
– John Bentin
Jan 28 at 19:23
add a comment |
1
$begingroup$
Please take a look at the Binomial Test.
$endgroup$
– denklo
Jan 28 at 11:26
$begingroup$
The question is wrong. It is no more answerable than "How long is a piece of string?". To make a sensible question, more information is needed. If we know the prior probability that the coin is biased, then we can calculate a posterior probability of it being so from the result of any given number of flips.
$endgroup$
– John Bentin
Jan 28 at 12:18
$begingroup$
@JohnBentin, I took a look at the binomial test. Isn't it correct to have a null hypothesis that the coin is biased such that $p>0.5 +epsilon$ and the alternative hypothesis that it is biased but with $pleq 0.5 +epsilon$? I wish to accept/reject the null hypothesis. Is this a well defined question?
$endgroup$
– user1936752
Jan 28 at 13:43
$begingroup$
The question didn't make clear that the set-up was to test the null hypothesis $p>0.5+epsilon$ versus the opposite. For any $pin[0,Bbb,, 1]$, you can calculate the probability of the outcome being as extreme as, or more extreme than, what is observed given that the probability of a head is $p$. But this looks like a pretty meaningless exercise. Even this useless calculation becomes impossible if we weaken the assumption from fixing P(head) to merely P(head) > $0.5+epsilon$ without specifying a distribution for P(head) in $[0.5+epsilon,Bbb, ,1]$.
$endgroup$
– John Bentin
Jan 28 at 19:23
1
1
$begingroup$
Please take a look at the Binomial Test.
$endgroup$
– denklo
Jan 28 at 11:26
$begingroup$
Please take a look at the Binomial Test.
$endgroup$
– denklo
Jan 28 at 11:26
$begingroup$
The question is wrong. It is no more answerable than "How long is a piece of string?". To make a sensible question, more information is needed. If we know the prior probability that the coin is biased, then we can calculate a posterior probability of it being so from the result of any given number of flips.
$endgroup$
– John Bentin
Jan 28 at 12:18
$begingroup$
The question is wrong. It is no more answerable than "How long is a piece of string?". To make a sensible question, more information is needed. If we know the prior probability that the coin is biased, then we can calculate a posterior probability of it being so from the result of any given number of flips.
$endgroup$
– John Bentin
Jan 28 at 12:18
$begingroup$
@JohnBentin, I took a look at the binomial test. Isn't it correct to have a null hypothesis that the coin is biased such that $p>0.5 +epsilon$ and the alternative hypothesis that it is biased but with $pleq 0.5 +epsilon$? I wish to accept/reject the null hypothesis. Is this a well defined question?
$endgroup$
– user1936752
Jan 28 at 13:43
$begingroup$
@JohnBentin, I took a look at the binomial test. Isn't it correct to have a null hypothesis that the coin is biased such that $p>0.5 +epsilon$ and the alternative hypothesis that it is biased but with $pleq 0.5 +epsilon$? I wish to accept/reject the null hypothesis. Is this a well defined question?
$endgroup$
– user1936752
Jan 28 at 13:43
$begingroup$
The question didn't make clear that the set-up was to test the null hypothesis $p>0.5+epsilon$ versus the opposite. For any $pin[0,Bbb,, 1]$, you can calculate the probability of the outcome being as extreme as, or more extreme than, what is observed given that the probability of a head is $p$. But this looks like a pretty meaningless exercise. Even this useless calculation becomes impossible if we weaken the assumption from fixing P(head) to merely P(head) > $0.5+epsilon$ without specifying a distribution for P(head) in $[0.5+epsilon,Bbb, ,1]$.
$endgroup$
– John Bentin
Jan 28 at 19:23
$begingroup$
The question didn't make clear that the set-up was to test the null hypothesis $p>0.5+epsilon$ versus the opposite. For any $pin[0,Bbb,, 1]$, you can calculate the probability of the outcome being as extreme as, or more extreme than, what is observed given that the probability of a head is $p$. But this looks like a pretty meaningless exercise. Even this useless calculation becomes impossible if we weaken the assumption from fixing P(head) to merely P(head) > $0.5+epsilon$ without specifying a distribution for P(head) in $[0.5+epsilon,Bbb, ,1]$.
$endgroup$
– John Bentin
Jan 28 at 19:23
add a comment |
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1
$begingroup$
Please take a look at the Binomial Test.
$endgroup$
– denklo
Jan 28 at 11:26
$begingroup$
The question is wrong. It is no more answerable than "How long is a piece of string?". To make a sensible question, more information is needed. If we know the prior probability that the coin is biased, then we can calculate a posterior probability of it being so from the result of any given number of flips.
$endgroup$
– John Bentin
Jan 28 at 12:18
$begingroup$
@JohnBentin, I took a look at the binomial test. Isn't it correct to have a null hypothesis that the coin is biased such that $p>0.5 +epsilon$ and the alternative hypothesis that it is biased but with $pleq 0.5 +epsilon$? I wish to accept/reject the null hypothesis. Is this a well defined question?
$endgroup$
– user1936752
Jan 28 at 13:43
$begingroup$
The question didn't make clear that the set-up was to test the null hypothesis $p>0.5+epsilon$ versus the opposite. For any $pin[0,Bbb,, 1]$, you can calculate the probability of the outcome being as extreme as, or more extreme than, what is observed given that the probability of a head is $p$. But this looks like a pretty meaningless exercise. Even this useless calculation becomes impossible if we weaken the assumption from fixing P(head) to merely P(head) > $0.5+epsilon$ without specifying a distribution for P(head) in $[0.5+epsilon,Bbb, ,1]$.
$endgroup$
– John Bentin
Jan 28 at 19:23