Can it be shown that $p(AB|C)=p(A|C)p(B|C)$ if $A$ and $B$ are independent events
up vote
0
down vote
favorite
It seems intuitive to me that $p(AB|C)=p(A|C)p(B|C)$ if $A$ and $B$ are independent events, but i cannot prove it. I have tried proving this using $p(AB)=p(A)p(B)$, Bayes' theorem and the definition of conditional probability, but to no avail. Can someone provide a proof or at least give me some directions on how to prove it (if it is true)? My interest in this is out of sheer curiosity.
probability independence conditional-probability
add a comment |
up vote
0
down vote
favorite
It seems intuitive to me that $p(AB|C)=p(A|C)p(B|C)$ if $A$ and $B$ are independent events, but i cannot prove it. I have tried proving this using $p(AB)=p(A)p(B)$, Bayes' theorem and the definition of conditional probability, but to no avail. Can someone provide a proof or at least give me some directions on how to prove it (if it is true)? My interest in this is out of sheer curiosity.
probability independence conditional-probability
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
It seems intuitive to me that $p(AB|C)=p(A|C)p(B|C)$ if $A$ and $B$ are independent events, but i cannot prove it. I have tried proving this using $p(AB)=p(A)p(B)$, Bayes' theorem and the definition of conditional probability, but to no avail. Can someone provide a proof or at least give me some directions on how to prove it (if it is true)? My interest in this is out of sheer curiosity.
probability independence conditional-probability
It seems intuitive to me that $p(AB|C)=p(A|C)p(B|C)$ if $A$ and $B$ are independent events, but i cannot prove it. I have tried proving this using $p(AB)=p(A)p(B)$, Bayes' theorem and the definition of conditional probability, but to no avail. Can someone provide a proof or at least give me some directions on how to prove it (if it is true)? My interest in this is out of sheer curiosity.
probability independence conditional-probability
probability independence conditional-probability
asked Nov 17 at 11:14
Haris Gusic
132
132
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
up vote
2
down vote
accepted
People often imagine this to be true but, alas, it is not. Here is a simple counterexample:
Suppose you toss a fair penny and a fair dime. Let $A$ be the event "the penny comes up $H$". Let $B$ be the event "the dime comes up $H$.". Let $C$ be the event "the two coins match".
Clearly, $A,B$ are independent.
However, they are not independent conditioned on $C$.
Indeed $$P(Acap B,|,C)=frac 12$$ but $$P(A,|,C)=frac 12=P(B,|,C)$$
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
accepted
People often imagine this to be true but, alas, it is not. Here is a simple counterexample:
Suppose you toss a fair penny and a fair dime. Let $A$ be the event "the penny comes up $H$". Let $B$ be the event "the dime comes up $H$.". Let $C$ be the event "the two coins match".
Clearly, $A,B$ are independent.
However, they are not independent conditioned on $C$.
Indeed $$P(Acap B,|,C)=frac 12$$ but $$P(A,|,C)=frac 12=P(B,|,C)$$
add a comment |
up vote
2
down vote
accepted
People often imagine this to be true but, alas, it is not. Here is a simple counterexample:
Suppose you toss a fair penny and a fair dime. Let $A$ be the event "the penny comes up $H$". Let $B$ be the event "the dime comes up $H$.". Let $C$ be the event "the two coins match".
Clearly, $A,B$ are independent.
However, they are not independent conditioned on $C$.
Indeed $$P(Acap B,|,C)=frac 12$$ but $$P(A,|,C)=frac 12=P(B,|,C)$$
add a comment |
up vote
2
down vote
accepted
up vote
2
down vote
accepted
People often imagine this to be true but, alas, it is not. Here is a simple counterexample:
Suppose you toss a fair penny and a fair dime. Let $A$ be the event "the penny comes up $H$". Let $B$ be the event "the dime comes up $H$.". Let $C$ be the event "the two coins match".
Clearly, $A,B$ are independent.
However, they are not independent conditioned on $C$.
Indeed $$P(Acap B,|,C)=frac 12$$ but $$P(A,|,C)=frac 12=P(B,|,C)$$
People often imagine this to be true but, alas, it is not. Here is a simple counterexample:
Suppose you toss a fair penny and a fair dime. Let $A$ be the event "the penny comes up $H$". Let $B$ be the event "the dime comes up $H$.". Let $C$ be the event "the two coins match".
Clearly, $A,B$ are independent.
However, they are not independent conditioned on $C$.
Indeed $$P(Acap B,|,C)=frac 12$$ but $$P(A,|,C)=frac 12=P(B,|,C)$$
edited Nov 17 at 11:26
answered Nov 17 at 11:19
lulu
38.2k24476
38.2k24476
add a comment |
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3002222%2fcan-it-be-shown-that-pabc-pacpbc-if-a-and-b-are-independent-event%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown