What is the conditional probability if we condition with the empty set?
It's an interesting question I think (or what is most likely I'm missing some subtle details). On one hand, we know that the empty set is independent of any other event so for any $A in mathcal{F}:$ $P( A , | , emptyset) = P(A)$, on the other hand, if we write this conditional probability using the definition, we might get that something is divided by zero. So what is $P( A , | , emptyset) :
\
P(A),, , 0 , ,text{or something undefined ?}$
p.s.: I added zero because intuitively if we know that everything is wrong (our condition) then A must be wrong (like never to happen).
probability probability-theory
asked Nov 25 '18 at 23:01
NickM
163
add a comment |
It's an interesting question I think (or what is most likely I'm missing some subtle details). On one hand, we know that the empty set is independent of any other event so for any $A in mathcal{F}:$ $P( A , | , emptyset) = P(A)$, on the other hand, if we write this conditional probability using the definition, we might get that something is divided by zero. So what is $P( A , | , emptyset) :
\
P(A),, , 0 , ,text{or something undefined ?}$
p.s.: I added zero because intuitively if we know that everything is wrong (our condition) then A must be wrong (like never to happen).
probability probability-theory
asked Nov 25 '18 at 23:01
NickM
163
2
The probability is not defined in such case.
– DavidPM
Nov 25 '18 at 23:02
How can you elaborate that?
– NickM
Nov 25 '18 at 23:02
1
Well, by definition, the conditional probability of A given B would be the probability of their intersection / conjunction divided by the probability of B. If B is the empty set you get null probability (the empty set has no measure) and thus would yield division by 0. Thus, undefined.
– DavidPM
Nov 25 '18 at 23:12
add a comment |
It's an interesting question I think (or what is most likely I'm missing some subtle details). On one hand, we know that the empty set is independent of any other event so for any $A in mathcal{F}:$ $P( A , | , emptyset) = P(A)$, on the other hand, if we write this conditional probability using the definition, we might get that something is divided by zero. So what is $P( A , | , emptyset) :
\
P(A),, , 0 , ,text{or something undefined ?}$
p.s.: I added zero because intuitively if we know that everything is wrong (our condition) then A must be wrong (like never to happen).
probability probability-theory
asked Nov 25 '18 at 23:01
NickM
163
It's an interesting question I think (or what is most likely I'm missing some subtle details). On one hand, we know that the empty set is independent of any other event so for any $A in mathcal{F}:$ $P( A , | , emptyset) = P(A)$, on the other hand, if we write this conditional probability using the definition, we might get that something is divided by zero. So what is $P( A , | , emptyset) :
\
P(A),, , 0 , ,text{or something undefined ?}$
p.s.: I added zero because intuitively if we know that everything is wrong (our condition) then A must be wrong (like never to happen).
probability probability-theory
probability probability-theory
asked Nov 25 '18 at 23:01
NickM
163
asked Nov 25 '18 at 23:01
NickM
163
edited Nov 25 '18 at 23:09
asked Nov 25 '18 at 23:01
NickM
163
asked Nov 25 '18 at 23:01
NickM
163
asked Nov 25 '18 at 23:01
NickM
163
163
2
The probability is not defined in such case.
– DavidPM
Nov 25 '18 at 23:02
How can you elaborate that?
– NickM
Nov 25 '18 at 23:02
1
Well, by definition, the conditional probability of A given B would be the probability of their intersection / conjunction divided by the probability of B. If B is the empty set you get null probability (the empty set has no measure) and thus would yield division by 0. Thus, undefined.
– DavidPM
Nov 25 '18 at 23:12
add a comment |
2
The probability is not defined in such case.
– DavidPM
Nov 25 '18 at 23:02
How can you elaborate that?
– NickM
Nov 25 '18 at 23:02
1
Well, by definition, the conditional probability of A given B would be the probability of their intersection / conjunction divided by the probability of B. If B is the empty set you get null probability (the empty set has no measure) and thus would yield division by 0. Thus, undefined.
– DavidPM
Nov 25 '18 at 23:12
2
2
The probability is not defined in such case.
– DavidPM
Nov 25 '18 at 23:02
The probability is not defined in such case.
– DavidPM
Nov 25 '18 at 23:02
How can you elaborate that?
– NickM
Nov 25 '18 at 23:02
How can you elaborate that?
– NickM
Nov 25 '18 at 23:02
1
1
Well, by definition, the conditional probability of A given B would be the probability of their intersection / conjunction divided by the probability of B. If B is the empty set you get null probability (the empty set has no measure) and thus would yield division by 0. Thus, undefined.
– DavidPM
Nov 25 '18 at 23:12
Well, by definition, the conditional probability of A given B would be the probability of their intersection / conjunction divided by the probability of B. If B is the empty set you get null probability (the empty set has no measure) and thus would yield division by 0. Thus, undefined.
– DavidPM
Nov 25 '18 at 23:12
add a comment |
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2
The probability is not defined in such case.
– DavidPM
Nov 25 '18 at 23:02
How can you elaborate that?
– NickM
Nov 25 '18 at 23:02
1
Well, by definition, the conditional probability of A given B would be the probability of their intersection / conjunction divided by the probability of B. If B is the empty set you get null probability (the empty set has no measure) and thus would yield division by 0. Thus, undefined.
– DavidPM
Nov 25 '18 at 23:12