Mixing
Event equaling the impossible event with probability $1$
$begingroup$
I have a question regarding events with probability $1$. On page 21 of Papoulis (Probability,Random Variables and Stochastic Process), there is an equation:
$P(A) = P(B) = P(AB)$
Now using this equation one conclusion has been derived that "If an event $N$ equals the impossible event with probability $1$ then $P(N)=0$. This does not, of course, mean that $N=phi$
Now my question why is $N$ not equal to $phi$ ?
Papoulis, A., Probability, random variables, and stochastic processes, New York: McGraw-Hill, XI, 583 p. (1965). ZBL0191.46704.
probability-theory
edited Feb 3 at 7:49
El borito
664216
asked Feb 3 at 6:31
Navneet DharNavneet Dhar
1
$endgroup$
add a comment |
$begingroup$
I have a question regarding events with probability $1$. On page 21 of Papoulis (Probability,Random Variables and Stochastic Process), there is an equation:
$P(A) = P(B) = P(AB)$
Now using this equation one conclusion has been derived that "If an event $N$ equals the impossible event with probability $1$ then $P(N)=0$. This does not, of course, mean that $N=phi$
Now my question why is $N$ not equal to $phi$ ?
Papoulis, A., Probability, random variables, and stochastic processes, New York: McGraw-Hill, XI, 583 p. (1965). ZBL0191.46704.
probability-theory
edited Feb 3 at 7:49
El borito
664216
asked Feb 3 at 6:31
Navneet DharNavneet Dhar
1
$endgroup$
1
$begingroup$
What is "phi"${}$?
$endgroup$
– Lord Shark the Unknown
Feb 3 at 6:48
$begingroup$
I think "phi" means the empty set.
$endgroup$
– Tori
Feb 3 at 7:43
$begingroup$
I would be curious to read how the OP understands the hypothesis that "an event A equals an event B with probability 1". I can think of a translation of this rather bizarre assertion into solid mathematics, but somehow, I doubt anyone asking the present question can reach it easily... Although I would love to be proven wrong. :-)
$endgroup$
– Did
Feb 3 at 8:32
add a comment |
$begingroup$
I have a question regarding events with probability $1$. On page 21 of Papoulis (Probability,Random Variables and Stochastic Process), there is an equation:
$P(A) = P(B) = P(AB)$
Now using this equation one conclusion has been derived that "If an event $N$ equals the impossible event with probability $1$ then $P(N)=0$. This does not, of course, mean that $N=phi$
Now my question why is $N$ not equal to $phi$ ?
Papoulis, A., Probability, random variables, and stochastic processes, New York: McGraw-Hill, XI, 583 p. (1965). ZBL0191.46704.
probability-theory
edited Feb 3 at 7:49
El borito
664216
asked Feb 3 at 6:31
Navneet DharNavneet Dhar
1
$endgroup$
I have a question regarding events with probability $1$. On page 21 of Papoulis (Probability,Random Variables and Stochastic Process), there is an equation:
$P(A) = P(B) = P(AB)$
Now using this equation one conclusion has been derived that "If an event $N$ equals the impossible event with probability $1$ then $P(N)=0$. This does not, of course, mean that $N=phi$
Now my question why is $N$ not equal to $phi$ ?
Papoulis, A., Probability, random variables, and stochastic processes, New York: McGraw-Hill, XI, 583 p. (1965). ZBL0191.46704.
probability-theory
probability-theory
edited Feb 3 at 7:49
El borito
664216
asked Feb 3 at 6:31
Navneet DharNavneet Dhar
1
edited Feb 3 at 7:49
El borito
664216
asked Feb 3 at 6:31
Navneet DharNavneet Dhar
1
edited Feb 3 at 7:49
El borito
664216
edited Feb 3 at 7:49
El borito
664216
edited Feb 3 at 7:49
El borito
664216
664216
asked Feb 3 at 6:31
Navneet DharNavneet Dhar
1
asked Feb 3 at 6:31
Navneet DharNavneet Dhar
1
asked Feb 3 at 6:31
Navneet DharNavneet Dhar
1
1
1
$begingroup$
What is "phi"${}$?
$endgroup$
– Lord Shark the Unknown
Feb 3 at 6:48
$begingroup$
I think "phi" means the empty set.
$endgroup$
– Tori
Feb 3 at 7:43
$begingroup$
I would be curious to read how the OP understands the hypothesis that "an event A equals an event B with probability 1". I can think of a translation of this rather bizarre assertion into solid mathematics, but somehow, I doubt anyone asking the present question can reach it easily... Although I would love to be proven wrong. :-)
$endgroup$
– Did
Feb 3 at 8:32
add a comment |
1
$begingroup$
What is "phi"${}$?
$endgroup$
– Lord Shark the Unknown
Feb 3 at 6:48
$begingroup$
I think "phi" means the empty set.
$endgroup$
– Tori
Feb 3 at 7:43
$begingroup$
I would be curious to read how the OP understands the hypothesis that "an event A equals an event B with probability 1". I can think of a translation of this rather bizarre assertion into solid mathematics, but somehow, I doubt anyone asking the present question can reach it easily... Although I would love to be proven wrong. :-)
$endgroup$
– Did
Feb 3 at 8:32
1
1
$begingroup$
What is "phi"${}$?
$endgroup$
– Lord Shark the Unknown
Feb 3 at 6:48
$begingroup$
What is "phi"${}$?
$endgroup$
– Lord Shark the Unknown
Feb 3 at 6:48
$begingroup$
I think "phi" means the empty set.
$endgroup$
– Tori
Feb 3 at 7:43
$begingroup$
I think "phi" means the empty set.
$endgroup$
– Tori
Feb 3 at 7:43
$begingroup$
I would be curious to read how the OP understands the hypothesis that "an event A equals an event B with probability 1". I can think of a translation of this rather bizarre assertion into solid mathematics, but somehow, I doubt anyone asking the present question can reach it easily... Although I would love to be proven wrong. :-)
$endgroup$
– Did
Feb 3 at 8:32
$begingroup$
I would be curious to read how the OP understands the hypothesis that "an event A equals an event B with probability 1". I can think of a translation of this rather bizarre assertion into solid mathematics, but somehow, I doubt anyone asking the present question can reach it easily... Although I would love to be proven wrong. :-)
$endgroup$
– Did
Feb 3 at 8:32
add a comment |
1 Answer
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$begingroup$
Your question is basically asking: If an event $N$ has probability 0, then why can't we conclude that $N = {varnothing}$? Now, note that ${varnothing} not= varnothing$. In other words, ${varnothing}$ is $underline{not}$ the empty set. (I think that you probability meant to ask why we can't conclude that $N = varnothing$.)
Anyhow, one of the three axioms of probability states:
$P(varnothing) = 0$,
however, it does not state that $P(A) = 0 Leftrightarrow A = varnothing$.
answered Feb 3 at 7:54
ToriTori
936
$endgroup$
add a comment |
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$begingroup$
Your question is basically asking: If an event $N$ has probability 0, then why can't we conclude that $N = {varnothing}$? Now, note that ${varnothing} not= varnothing$. In other words, ${varnothing}$ is $underline{not}$ the empty set. (I think that you probability meant to ask why we can't conclude that $N = varnothing$.)
Anyhow, one of the three axioms of probability states:
$P(varnothing) = 0$,
however, it does not state that $P(A) = 0 Leftrightarrow A = varnothing$.
answered Feb 3 at 7:54
ToriTori
936
$endgroup$
add a comment |
$begingroup$
Your question is basically asking: If an event $N$ has probability 0, then why can't we conclude that $N = {varnothing}$? Now, note that ${varnothing} not= varnothing$. In other words, ${varnothing}$ is $underline{not}$ the empty set. (I think that you probability meant to ask why we can't conclude that $N = varnothing$.)
Anyhow, one of the three axioms of probability states:
$P(varnothing) = 0$,
however, it does not state that $P(A) = 0 Leftrightarrow A = varnothing$.
answered Feb 3 at 7:54
ToriTori
936
$endgroup$
add a comment |
$begingroup$
Your question is basically asking: If an event $N$ has probability 0, then why can't we conclude that $N = {varnothing}$? Now, note that ${varnothing} not= varnothing$. In other words, ${varnothing}$ is $underline{not}$ the empty set. (I think that you probability meant to ask why we can't conclude that $N = varnothing$.)
Anyhow, one of the three axioms of probability states:
$P(varnothing) = 0$,
however, it does not state that $P(A) = 0 Leftrightarrow A = varnothing$.
answered Feb 3 at 7:54
ToriTori
936
$endgroup$
Your question is basically asking: If an event $N$ has probability 0, then why can't we conclude that $N = {varnothing}$? Now, note that ${varnothing} not= varnothing$. In other words, ${varnothing}$ is $underline{not}$ the empty set. (I think that you probability meant to ask why we can't conclude that $N = varnothing$.)
Anyhow, one of the three axioms of probability states:
$P(varnothing) = 0$,
however, it does not state that $P(A) = 0 Leftrightarrow A = varnothing$.
answered Feb 3 at 7:54
ToriTori
936
answered Feb 3 at 7:54
ToriTori
936
answered Feb 3 at 7:54
ToriTori
936
answered Feb 3 at 7:54
ToriTori
936
936
add a comment |
add a comment |
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1
$begingroup$
What is "phi"${}$?
$endgroup$
– Lord Shark the Unknown
Feb 3 at 6:48
$begingroup$
I think "phi" means the empty set.
$endgroup$
– Tori
Feb 3 at 7:43
$begingroup$
I would be curious to read how the OP understands the hypothesis that "an event A equals an event B with probability 1". I can think of a translation of this rather bizarre assertion into solid mathematics, but somehow, I doubt anyone asking the present question can reach it easily... Although I would love to be proven wrong. :-)
$endgroup$
– Did
Feb 3 at 8:32