Definition of dependence in probability

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Here is classical definition and example of dependent events.

"When two events are said to be dependent, the probability of one event occurring influences the likelihood of the other event. For example, if you were to draw a two cards from a deck of $52$ cards. If on your first draw you had an ace and you put that aside, the probability of drawing an ace on the second draw is greatly changed because you drew an ace the first time".

Let’s consider another scenario: Suppose I apply to a job. There are two interviews. The second interview ($B$) will take place only if I pass the first interview ($A$). So, we have probabilities of $P(A)$ and $P(B)$. Two events per se do not depend on each other because different people conduct the interviews. But $B$ will not take place if $A$ was a failure. So $P(B)ne P(Bmid A)$. So, can it be said that events $A$ and $B$ are dependent?

Thanks!

probability definition independence

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edited Aug 6 at 11:00

asked Aug 6 at 8:09

John

463

• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  Your title "combinatorics problem" does not match the question or tags at all.
  – Servaes
  Aug 6 at 8:12
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  And of course the two events depend on eachother; you say yourself that the second interview will take place only if you pass the first. Perhaps you should clarify (for yourself) what $A$ and $B$ are; the interviews taking place, or passing the interviews?
  – Servaes
  Aug 6 at 8:12
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Thanks. Sorry for the title confusion. I fixed it. So, if I define the event A as "Passing the first interview" and event B as "Passing the second interview", then they are independent. But if it is "passing" each stage, then they are dependent. Correct?
  – John
  Aug 6 at 8:24
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @John, no, you cannot pass an interview if the interview does not take place.
  – Graham Kemp
  Aug 6 at 8:38
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  OK, OK. I see. Now it is clear. Thanks
  – John
  Aug 6 at 8:41
  

  
  
  
  

add a comment | 

up vote
2
down vote

favorite

1

Here is classical definition and example of dependent events.

"When two events are said to be dependent, the probability of one event occurring influences the likelihood of the other event. For example, if you were to draw a two cards from a deck of $52$ cards. If on your first draw you had an ace and you put that aside, the probability of drawing an ace on the second draw is greatly changed because you drew an ace the first time".

Let’s consider another scenario: Suppose I apply to a job. There are two interviews. The second interview ($B$) will take place only if I pass the first interview ($A$). So, we have probabilities of $P(A)$ and $P(B)$. Two events per se do not depend on each other because different people conduct the interviews. But $B$ will not take place if $A$ was a failure. So $P(B)ne P(Bmid A)$. So, can it be said that events $A$ and $B$ are dependent?

Thanks!

probability definition independence

share|cite|improve this question

edited Aug 6 at 11:00

asked Aug 6 at 8:09

John

463

• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  Your title "combinatorics problem" does not match the question or tags at all.
  – Servaes
  Aug 6 at 8:12
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  And of course the two events depend on eachother; you say yourself that the second interview will take place only if you pass the first. Perhaps you should clarify (for yourself) what $A$ and $B$ are; the interviews taking place, or passing the interviews?
  – Servaes
  Aug 6 at 8:12
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Thanks. Sorry for the title confusion. I fixed it. So, if I define the event A as "Passing the first interview" and event B as "Passing the second interview", then they are independent. But if it is "passing" each stage, then they are dependent. Correct?
  – John
  Aug 6 at 8:24
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @John, no, you cannot pass an interview if the interview does not take place.
  – Graham Kemp
  Aug 6 at 8:38
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  OK, OK. I see. Now it is clear. Thanks
  – John
  Aug 6 at 8:41
  

  
  
  
  

add a comment | 

up vote
2
down vote

favorite

1

up vote
2
down vote

favorite

1

1

Here is classical definition and example of dependent events.

"When two events are said to be dependent, the probability of one event occurring influences the likelihood of the other event. For example, if you were to draw a two cards from a deck of $52$ cards. If on your first draw you had an ace and you put that aside, the probability of drawing an ace on the second draw is greatly changed because you drew an ace the first time".

Let’s consider another scenario: Suppose I apply to a job. There are two interviews. The second interview ($B$) will take place only if I pass the first interview ($A$). So, we have probabilities of $P(A)$ and $P(B)$. Two events per se do not depend on each other because different people conduct the interviews. But $B$ will not take place if $A$ was a failure. So $P(B)ne P(Bmid A)$. So, can it be said that events $A$ and $B$ are dependent?

Thanks!

probability definition independence

share|cite|improve this question

edited Aug 6 at 11:00

asked Aug 6 at 8:09

John

463

Here is classical definition and example of dependent events.

"When two events are said to be dependent, the probability of one event occurring influences the likelihood of the other event. For example, if you were to draw a two cards from a deck of $52$ cards. If on your first draw you had an ace and you put that aside, the probability of drawing an ace on the second draw is greatly changed because you drew an ace the first time".

Let’s consider another scenario: Suppose I apply to a job. There are two interviews. The second interview ($B$) will take place only if I pass the first interview ($A$). So, we have probabilities of $P(A)$ and $P(B)$. Two events per se do not depend on each other because different people conduct the interviews. But $B$ will not take place if $A$ was a failure. So $P(B)ne P(Bmid A)$. So, can it be said that events $A$ and $B$ are dependent?

Thanks!

probability definition independence

share|cite|improve this question

edited Aug 6 at 11:00

asked Aug 6 at 8:09

John

463

share|cite|improve this question

share|cite|improve this question

edited Aug 6 at 11:00

edited Aug 6 at 11:00

edited Aug 6 at 11:00

asked Aug 6 at 8:09

John

463

asked Aug 6 at 8:09

John

463

asked Aug 6 at 8:09

John

463

463

• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  Your title "combinatorics problem" does not match the question or tags at all.
  – Servaes
  Aug 6 at 8:12
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  And of course the two events depend on eachother; you say yourself that the second interview will take place only if you pass the first. Perhaps you should clarify (for yourself) what $A$ and $B$ are; the interviews taking place, or passing the interviews?
  – Servaes
  Aug 6 at 8:12
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Thanks. Sorry for the title confusion. I fixed it. So, if I define the event A as "Passing the first interview" and event B as "Passing the second interview", then they are independent. But if it is "passing" each stage, then they are dependent. Correct?
  – John
  Aug 6 at 8:24
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @John, no, you cannot pass an interview if the interview does not take place.
  – Graham Kemp
  Aug 6 at 8:38
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  OK, OK. I see. Now it is clear. Thanks
  – John
  Aug 6 at 8:41
  

  
  
  
  

add a comment | 

• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  Your title "combinatorics problem" does not match the question or tags at all.
  – Servaes
  Aug 6 at 8:12
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  And of course the two events depend on eachother; you say yourself that the second interview will take place only if you pass the first. Perhaps you should clarify (for yourself) what $A$ and $B$ are; the interviews taking place, or passing the interviews?
  – Servaes
  Aug 6 at 8:12
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Thanks. Sorry for the title confusion. I fixed it. So, if I define the event A as "Passing the first interview" and event B as "Passing the second interview", then they are independent. But if it is "passing" each stage, then they are dependent. Correct?
  – John
  Aug 6 at 8:24
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @John, no, you cannot pass an interview if the interview does not take place.
  – Graham Kemp
  Aug 6 at 8:38
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  OK, OK. I see. Now it is clear. Thanks
  – John
  Aug 6 at 8:41
  

  
  
  
  

2

2

Your title "combinatorics problem" does not match the question or tags at all.
– Servaes
Aug 6 at 8:12

Your title "combinatorics problem" does not match the question or tags at all.
– Servaes
Aug 6 at 8:12

1

1

And of course the two events depend on eachother; you say yourself that the second interview will take place only if you pass the first. Perhaps you should clarify (for yourself) what $A$ and $B$ are; the interviews taking place, or passing the interviews?
– Servaes
Aug 6 at 8:12

And of course the two events depend on eachother; you say yourself that the second interview will take place only if you pass the first. Perhaps you should clarify (for yourself) what $A$ and $B$ are; the interviews taking place, or passing the interviews?
– Servaes
Aug 6 at 8:12

Thanks. Sorry for the title confusion. I fixed it. So, if I define the event A as "Passing the first interview" and event B as "Passing the second interview", then they are independent. But if it is "passing" each stage, then they are dependent. Correct?
– John
Aug 6 at 8:24

Thanks. Sorry for the title confusion. I fixed it. So, if I define the event A as "Passing the first interview" and event B as "Passing the second interview", then they are independent. But if it is "passing" each stage, then they are dependent. Correct?
– John
Aug 6 at 8:24

@John, no, you cannot pass an interview if the interview does not take place.
– Graham Kemp
Aug 6 at 8:38

@John, no, you cannot pass an interview if the interview does not take place.
– Graham Kemp
Aug 6 at 8:38

OK, OK. I see. Now it is clear. Thanks
– John
Aug 6 at 8:41

OK, OK. I see. Now it is clear. Thanks
– John
Aug 6 at 8:41

add a comment | 

3 Answers
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up vote
1
down vote

Perhaps the most concise definition of independent events is $P(A,textand,B)=P(A)P(B)$. In your example, constants $p,,q$ exist for which $P(A)=p,,P(A,textand,B)=P(B)=q$, with independence only if $p=1$.

share|cite|improve this answer

answered Aug 6 at 8:44

J.G.

13.4k11424

add a comment | 

up vote
0
down vote

1. "When two events are said to be dependent, the probability of one event occurring influences the likelihood of the other event."




2. "Let’s consider another scenario: Suppose I apply to a job. There are two interviews. The second interview (B) will take place only if I pass the first interview (A). So, we have probabilities of P(A) and P(B). Two events per se do not depend on each other because different people conduct the interviews. But B will not take place if A was a failure."

Reasoning: Event B cannot occur without event A occurring before it.

Conclusion: A and B are dependent events.

share|cite|improve this answer

answered Aug 6 at 8:18

CaptainAmerica16

171110

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  Thanks. Will you agree that this is the matter of event definition (see my reply to Servaes)?
  – John
  Aug 6 at 8:26
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  It definitely is a matter of definition, but it also is a matter of phrasing. Reading your comment, each definition effectively means the same thing if you are defining Event A as a parameter of getting to Event B like you have above.
  – CaptainAmerica16
  Aug 6 at 8:31
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  To be truly independent, each event cannot have an effect on the outcome of the other.
  – CaptainAmerica16
  Aug 6 at 8:32
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Think of A and B as two separate interviews. Passing A will not result in you gaining or losing B unless of course you add more parameters such as the time of the interviews.
  – CaptainAmerica16
  Aug 6 at 8:35
  
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  Thanks for the help!
  – John
  Aug 6 at 8:41
  

  
  
  
  

 | 
show 1 more comment

up vote
0
down vote

We say two events $A$ and $B$ are independent if knowing that one had occurred gave us no information about whether the other had occurred is

$$P(A|B) = P(A) $$

and

$$P(B|A) = P(B)$$

now we know

$$ P(A) = P(A|B) = fracP(A cap B)P(B) $$

then

$$ P(A cap B) = P(A)P(B)$$

clear if they do depend on each other this is not the case

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answered Aug 6 at 8:40

RHowe

1,025815

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3 Answers
3

active

oldest

votes

3 Answers
3

active

oldest

votes

active

oldest

votes

active

oldest

votes

up vote
1
down vote

Perhaps the most concise definition of independent events is $P(A,textand,B)=P(A)P(B)$. In your example, constants $p,,q$ exist for which $P(A)=p,,P(A,textand,B)=P(B)=q$, with independence only if $p=1$.

share|cite|improve this answer

answered Aug 6 at 8:44

J.G.

13.4k11424

add a comment | 

up vote
1
down vote

Perhaps the most concise definition of independent events is $P(A,textand,B)=P(A)P(B)$. In your example, constants $p,,q$ exist for which $P(A)=p,,P(A,textand,B)=P(B)=q$, with independence only if $p=1$.

share|cite|improve this answer

answered Aug 6 at 8:44

J.G.

13.4k11424

add a comment | 

up vote
1
down vote

up vote
1
down vote

Perhaps the most concise definition of independent events is $P(A,textand,B)=P(A)P(B)$. In your example, constants $p,,q$ exist for which $P(A)=p,,P(A,textand,B)=P(B)=q$, with independence only if $p=1$.

share|cite|improve this answer

answered Aug 6 at 8:44

J.G.

13.4k11424

Perhaps the most concise definition of independent events is $P(A,textand,B)=P(A)P(B)$. In your example, constants $p,,q$ exist for which $P(A)=p,,P(A,textand,B)=P(B)=q$, with independence only if $p=1$.

share|cite|improve this answer

answered Aug 6 at 8:44

J.G.

13.4k11424

share|cite|improve this answer

share|cite|improve this answer

answered Aug 6 at 8:44

J.G.

13.4k11424

answered Aug 6 at 8:44

J.G.

13.4k11424

answered Aug 6 at 8:44

J.G.

13.4k11424

13.4k11424

add a comment | 

add a comment | 

up vote
0
down vote

1. "When two events are said to be dependent, the probability of one event occurring influences the likelihood of the other event."




2. "Let’s consider another scenario: Suppose I apply to a job. There are two interviews. The second interview (B) will take place only if I pass the first interview (A). So, we have probabilities of P(A) and P(B). Two events per se do not depend on each other because different people conduct the interviews. But B will not take place if A was a failure."

Reasoning: Event B cannot occur without event A occurring before it.

Conclusion: A and B are dependent events.

share|cite|improve this answer

answered Aug 6 at 8:18

CaptainAmerica16

171110

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  Thanks. Will you agree that this is the matter of event definition (see my reply to Servaes)?
  – John
  Aug 6 at 8:26
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  It definitely is a matter of definition, but it also is a matter of phrasing. Reading your comment, each definition effectively means the same thing if you are defining Event A as a parameter of getting to Event B like you have above.
  – CaptainAmerica16
  Aug 6 at 8:31
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  To be truly independent, each event cannot have an effect on the outcome of the other.
  – CaptainAmerica16
  Aug 6 at 8:32
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Think of A and B as two separate interviews. Passing A will not result in you gaining or losing B unless of course you add more parameters such as the time of the interviews.
  – CaptainAmerica16
  Aug 6 at 8:35
  
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  Thanks for the help!
  – John
  Aug 6 at 8:41
  

  
  
  
  

 | 
show 1 more comment

up vote
0
down vote

1. "When two events are said to be dependent, the probability of one event occurring influences the likelihood of the other event."




2. "Let’s consider another scenario: Suppose I apply to a job. There are two interviews. The second interview (B) will take place only if I pass the first interview (A). So, we have probabilities of P(A) and P(B). Two events per se do not depend on each other because different people conduct the interviews. But B will not take place if A was a failure."

Reasoning: Event B cannot occur without event A occurring before it.

Conclusion: A and B are dependent events.

share|cite|improve this answer

answered Aug 6 at 8:18

CaptainAmerica16

171110

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  Thanks. Will you agree that this is the matter of event definition (see my reply to Servaes)?
  – John
  Aug 6 at 8:26
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  It definitely is a matter of definition, but it also is a matter of phrasing. Reading your comment, each definition effectively means the same thing if you are defining Event A as a parameter of getting to Event B like you have above.
  – CaptainAmerica16
  Aug 6 at 8:31
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  To be truly independent, each event cannot have an effect on the outcome of the other.
  – CaptainAmerica16
  Aug 6 at 8:32
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Think of A and B as two separate interviews. Passing A will not result in you gaining or losing B unless of course you add more parameters such as the time of the interviews.
  – CaptainAmerica16
  Aug 6 at 8:35
  
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  Thanks for the help!
  – John
  Aug 6 at 8:41
  

  
  
  
  

 | 
show 1 more comment

up vote
0
down vote

up vote
0
down vote

1. "When two events are said to be dependent, the probability of one event occurring influences the likelihood of the other event."




2. "Let’s consider another scenario: Suppose I apply to a job. There are two interviews. The second interview (B) will take place only if I pass the first interview (A). So, we have probabilities of P(A) and P(B). Two events per se do not depend on each other because different people conduct the interviews. But B will not take place if A was a failure."

Reasoning: Event B cannot occur without event A occurring before it.

Conclusion: A and B are dependent events.

share|cite|improve this answer

answered Aug 6 at 8:18

CaptainAmerica16

171110

1. "When two events are said to be dependent, the probability of one event occurring influences the likelihood of the other event."




2. "Let’s consider another scenario: Suppose I apply to a job. There are two interviews. The second interview (B) will take place only if I pass the first interview (A). So, we have probabilities of P(A) and P(B). Two events per se do not depend on each other because different people conduct the interviews. But B will not take place if A was a failure."

Reasoning: Event B cannot occur without event A occurring before it.

Conclusion: A and B are dependent events.

share|cite|improve this answer

answered Aug 6 at 8:18

CaptainAmerica16

171110

share|cite|improve this answer

share|cite|improve this answer

answered Aug 6 at 8:18

CaptainAmerica16

171110

answered Aug 6 at 8:18

CaptainAmerica16

171110

answered Aug 6 at 8:18

CaptainAmerica16

171110

171110

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  Thanks. Will you agree that this is the matter of event definition (see my reply to Servaes)?
  – John
  Aug 6 at 8:26
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  It definitely is a matter of definition, but it also is a matter of phrasing. Reading your comment, each definition effectively means the same thing if you are defining Event A as a parameter of getting to Event B like you have above.
  – CaptainAmerica16
  Aug 6 at 8:31
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  To be truly independent, each event cannot have an effect on the outcome of the other.
  – CaptainAmerica16
  Aug 6 at 8:32
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Think of A and B as two separate interviews. Passing A will not result in you gaining or losing B unless of course you add more parameters such as the time of the interviews.
  – CaptainAmerica16
  Aug 6 at 8:35
  
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  Thanks for the help!
  – John
  Aug 6 at 8:41
  

  
  
  
  

 | 
show 1 more comment

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  Thanks. Will you agree that this is the matter of event definition (see my reply to Servaes)?
  – John
  Aug 6 at 8:26
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  It definitely is a matter of definition, but it also is a matter of phrasing. Reading your comment, each definition effectively means the same thing if you are defining Event A as a parameter of getting to Event B like you have above.
  – CaptainAmerica16
  Aug 6 at 8:31
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  To be truly independent, each event cannot have an effect on the outcome of the other.
  – CaptainAmerica16
  Aug 6 at 8:32
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  Think of A and B as two separate interviews. Passing A will not result in you gaining or losing B unless of course you add more parameters such as the time of the interviews.
  – CaptainAmerica16
  Aug 6 at 8:35
  
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  Thanks for the help!
  – John
  Aug 6 at 8:41
  

  
  
  
  

1

1

Thanks. Will you agree that this is the matter of event definition (see my reply to Servaes)?
– John
Aug 6 at 8:26

Thanks. Will you agree that this is the matter of event definition (see my reply to Servaes)?
– John
Aug 6 at 8:26

It definitely is a matter of definition, but it also is a matter of phrasing. Reading your comment, each definition effectively means the same thing if you are defining Event A as a parameter of getting to Event B like you have above.
– CaptainAmerica16
Aug 6 at 8:31

It definitely is a matter of definition, but it also is a matter of phrasing. Reading your comment, each definition effectively means the same thing if you are defining Event A as a parameter of getting to Event B like you have above.
– CaptainAmerica16
Aug 6 at 8:31

To be truly independent, each event cannot have an effect on the outcome of the other.
– CaptainAmerica16
Aug 6 at 8:32

To be truly independent, each event cannot have an effect on the outcome of the other.
– CaptainAmerica16
Aug 6 at 8:32

Think of A and B as two separate interviews. Passing A will not result in you gaining or losing B unless of course you add more parameters such as the time of the interviews.
– CaptainAmerica16
Aug 6 at 8:35

Think of A and B as two separate interviews. Passing A will not result in you gaining or losing B unless of course you add more parameters such as the time of the interviews.
– CaptainAmerica16
Aug 6 at 8:35

1

1

Thanks for the help!
– John
Aug 6 at 8:41

Thanks for the help!
– John
Aug 6 at 8:41

 | 
show 1 more comment

up vote
0
down vote

We say two events $A$ and $B$ are independent if knowing that one had occurred gave us no information about whether the other had occurred is

$$P(A|B) = P(A) $$

and

$$P(B|A) = P(B)$$

now we know

$$ P(A) = P(A|B) = fracP(A cap B)P(B) $$

then

$$ P(A cap B) = P(A)P(B)$$

clear if they do depend on each other this is not the case

share|cite|improve this answer

answered Aug 6 at 8:40

RHowe

1,025815

add a comment | 

up vote
0
down vote

We say two events $A$ and $B$ are independent if knowing that one had occurred gave us no information about whether the other had occurred is

$$P(A|B) = P(A) $$

and

$$P(B|A) = P(B)$$

now we know

$$ P(A) = P(A|B) = fracP(A cap B)P(B) $$

then

$$ P(A cap B) = P(A)P(B)$$

clear if they do depend on each other this is not the case

share|cite|improve this answer

answered Aug 6 at 8:40

RHowe

1,025815

add a comment | 

up vote
0
down vote

up vote
0
down vote

We say two events $A$ and $B$ are independent if knowing that one had occurred gave us no information about whether the other had occurred is

$$P(A|B) = P(A) $$

and

$$P(B|A) = P(B)$$

now we know

$$ P(A) = P(A|B) = fracP(A cap B)P(B) $$

then

$$ P(A cap B) = P(A)P(B)$$

clear if they do depend on each other this is not the case

share|cite|improve this answer

answered Aug 6 at 8:40

RHowe

1,025815

We say two events $A$ and $B$ are independent if knowing that one had occurred gave us no information about whether the other had occurred is

$$P(A|B) = P(A) $$

and

$$P(B|A) = P(B)$$

now we know

$$ P(A) = P(A|B) = fracP(A cap B)P(B) $$

then

$$ P(A cap B) = P(A)P(B)$$

clear if they do depend on each other this is not the case

share|cite|improve this answer

answered Aug 6 at 8:40

RHowe

1,025815

share|cite|improve this answer

share|cite|improve this answer

answered Aug 6 at 8:40

RHowe

1,025815

answered Aug 6 at 8:40

RHowe

1,025815

answered Aug 6 at 8:40

RHowe

1,025815

1,025815

add a comment | 

add a comment | 

 

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