Prooving the Independence of two events
Let $A$ be event and probability $mathbb{P}(A)$ is $0$ or $1$. How to show that two events $A$ and $B$ are independent of each other. Here $B$ is any other event.
So I think I need to proove $mathbb{P}(B|A)= mathbb{P}(A)$ or what?. How to start and what to do?
probability probability-theory independence
asked Dec 3 '18 at 8:14
Atstovas
697
add a comment |
Let $A$ be event and probability $mathbb{P}(A)$ is $0$ or $1$. How to show that two events $A$ and $B$ are independent of each other. Here $B$ is any other event.
So I think I need to proove $mathbb{P}(B|A)= mathbb{P}(A)$ or what?. How to start and what to do?
probability probability-theory independence
asked Dec 3 '18 at 8:14
Atstovas
697
add a comment |
Let $A$ be event and probability $mathbb{P}(A)$ is $0$ or $1$. How to show that two events $A$ and $B$ are independent of each other. Here $B$ is any other event.
So I think I need to proove $mathbb{P}(B|A)= mathbb{P}(A)$ or what?. How to start and what to do?
probability probability-theory independence
asked Dec 3 '18 at 8:14
Atstovas
697
Let $A$ be event and probability $mathbb{P}(A)$ is $0$ or $1$. How to show that two events $A$ and $B$ are independent of each other. Here $B$ is any other event.
So I think I need to proove $mathbb{P}(B|A)= mathbb{P}(A)$ or what?. How to start and what to do?
probability probability-theory independence
probability probability-theory independence
asked Dec 3 '18 at 8:14
Atstovas
697
asked Dec 3 '18 at 8:14
Atstovas
697
asked Dec 3 '18 at 8:14
Atstovas
697
asked Dec 3 '18 at 8:14
Atstovas
697
asked Dec 3 '18 at 8:14
Atstovas
697
697
add a comment |
add a comment |
1 Answer
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If $P(A)=0$ then $P(A)P(B)=0$ and $P(Acap B)leq P(A)=0$ so $P(Acap B)=P(A)P(B)$. If $P(A)=1$ then $P(A)P(B)=P(B)$ and $P(Acap B)=P(B)$ becasue $P(B)=P(A cap B)+P(Bsetminus A)$ and the second term is $0$. [ $P(Bsetminus A)leq P(A^{c})=1-P(A)=0$].
answered Dec 3 '18 at 8:21
Kavi Rama Murthy
50.4k31854
What does $A^C$ mean?
– Atstovas
Dec 5 '18 at 10:24
@Atstovas $A^{c}$ is the compliment of $A$.
– Kavi Rama Murthy
Dec 5 '18 at 10:25
add a comment |
Your Answer
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1 Answer
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active
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1 Answer
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active
oldest
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active
oldest
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active
oldest
votes
If $P(A)=0$ then $P(A)P(B)=0$ and $P(Acap B)leq P(A)=0$ so $P(Acap B)=P(A)P(B)$. If $P(A)=1$ then $P(A)P(B)=P(B)$ and $P(Acap B)=P(B)$ becasue $P(B)=P(A cap B)+P(Bsetminus A)$ and the second term is $0$. [ $P(Bsetminus A)leq P(A^{c})=1-P(A)=0$].
answered Dec 3 '18 at 8:21
Kavi Rama Murthy
50.4k31854
What does $A^C$ mean?
– Atstovas
Dec 5 '18 at 10:24
@Atstovas $A^{c}$ is the compliment of $A$.
– Kavi Rama Murthy
Dec 5 '18 at 10:25
add a comment |
If $P(A)=0$ then $P(A)P(B)=0$ and $P(Acap B)leq P(A)=0$ so $P(Acap B)=P(A)P(B)$. If $P(A)=1$ then $P(A)P(B)=P(B)$ and $P(Acap B)=P(B)$ becasue $P(B)=P(A cap B)+P(Bsetminus A)$ and the second term is $0$. [ $P(Bsetminus A)leq P(A^{c})=1-P(A)=0$].
answered Dec 3 '18 at 8:21
Kavi Rama Murthy
50.4k31854
What does $A^C$ mean?
– Atstovas
Dec 5 '18 at 10:24
@Atstovas $A^{c}$ is the compliment of $A$.
– Kavi Rama Murthy
Dec 5 '18 at 10:25
add a comment |
If $P(A)=0$ then $P(A)P(B)=0$ and $P(Acap B)leq P(A)=0$ so $P(Acap B)=P(A)P(B)$. If $P(A)=1$ then $P(A)P(B)=P(B)$ and $P(Acap B)=P(B)$ becasue $P(B)=P(A cap B)+P(Bsetminus A)$ and the second term is $0$. [ $P(Bsetminus A)leq P(A^{c})=1-P(A)=0$].
answered Dec 3 '18 at 8:21
Kavi Rama Murthy
50.4k31854
If $P(A)=0$ then $P(A)P(B)=0$ and $P(Acap B)leq P(A)=0$ so $P(Acap B)=P(A)P(B)$. If $P(A)=1$ then $P(A)P(B)=P(B)$ and $P(Acap B)=P(B)$ becasue $P(B)=P(A cap B)+P(Bsetminus A)$ and the second term is $0$. [ $P(Bsetminus A)leq P(A^{c})=1-P(A)=0$].
answered Dec 3 '18 at 8:21
Kavi Rama Murthy
50.4k31854
answered Dec 3 '18 at 8:21
Kavi Rama Murthy
50.4k31854
answered Dec 3 '18 at 8:21
Kavi Rama Murthy
50.4k31854
answered Dec 3 '18 at 8:21
Kavi Rama Murthy
50.4k31854
50.4k31854
What does $A^C$ mean?
– Atstovas
Dec 5 '18 at 10:24
@Atstovas $A^{c}$ is the compliment of $A$.
– Kavi Rama Murthy
Dec 5 '18 at 10:25
add a comment |
What does $A^C$ mean?
– Atstovas
Dec 5 '18 at 10:24
@Atstovas $A^{c}$ is the compliment of $A$.
– Kavi Rama Murthy
Dec 5 '18 at 10:25
What does $A^C$ mean?
– Atstovas
Dec 5 '18 at 10:24
What does $A^C$ mean?
– Atstovas
Dec 5 '18 at 10:24
@Atstovas $A^{c}$ is the compliment of $A$.
– Kavi Rama Murthy
Dec 5 '18 at 10:25
@Atstovas $A^{c}$ is the compliment of $A$.
– Kavi Rama Murthy
Dec 5 '18 at 10:25
add a comment |
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