Conditional Probability

In summary, the discussion focuses on the conditional probability of event A given event A∩B and A∪B. It is mentioned that in the case of A and A∪B being independent, the probability of either event occurring is 1, which can also be achieved by setting P(A∩B)=1. However, there is another way to achieve this by setting P(A∪B)=1. The conversation also considers the case of B∩A=∅ and discusses the dependence or independence of two intersecting events, noting that it cannot be proved just by knowing the intersection is non-empty and additional assumptions are needed.
  • #1
PFuser1232
479
20
##P(A|A∩B) = \frac{P(A∩(A∩B))}{P(A∩B)} = \frac{P(A∩B)}{P(A∩B)} = 1##
So given the the event "A and B" as the sample space, the probability of A occurring is 1.

##P(A|A∪B) = \frac{P(A∩(A∪B))}{P(A∪B)} = \frac{P(A)}{P(A∪B)}##
Those two events are independent if and only if the probability of "A or B" occurring is 1, in which case the conditional probability of A equals the probability of A.

Is my reasoning correct?
 
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  • #2
you mean that A and A∪B are independent if P(A∪B)=1 ? yes, this is a special case of independent events. There is one other special case which can mean that A and A∪B are independent. You are trying to make the last equation go from
[tex]P(A|A∪B) = \frac{P(A)}{P(A∪B)}[/tex]
to become:
[tex]P(A|A∪B) = P(A)[/tex]
right? and you did this by setting P(A∪B)=1. But there is another way also.
 
  • #3
MohammedRady97 said:
##P(A|A∩B) = \frac{P(A∩(A∩B))}{P(A∩B)} = \frac{P(A∩B)}{P(A∩B)} = 1##
So given the the event "A and B" as the sample space, the probability of A occurring is 1.
We'd have to consider the case [itex] B \cap A = \emptyset [/itex]

##P(A|A∪B) = \frac{P(A∩(A∪B))}{P(A∪B)} = \frac{P(A)}{P(A∪B)}##
Those two events are independent
Which two events? [itex] A [/itex] and [itex] A \cup B [/itex] ?

if and only if the probability of "A or B" occurring is 1

Suppose [itex] B = \emptyset [/itex].

I think your results are interesting and worth perfecting by taking care of the exceptional cases.
 
  • #4
Stephen Tashi said:
We'd have to consider the case [itex] B \cap A = \emptyset [/itex]Which two events? [itex] A [/itex] and [itex] A \cup B [/itex] ?
Suppose [itex] B = \emptyset [/itex].

I think your results are interesting and worth perfecting by taking care of the exceptional cases.

What about ##P(A|B)##? My gut tells me that since A and B intersect, A and B must be dependent. This is wrong, of course, but how do I prove the dependence (or independence) of two intersecting events?
 
  • #5
MohammedRady97 said:
What about ##P(A|B)##?

I don't know what your are asking.

My gut tells me that since A and B intersect, A and B must be dependent. This is wrong, of course, but how do I prove the dependence (or independence) of two intersecting events?

You can't prove the dependence or independence of two events that have a non-empty intersection just from knowing the intersection is non-empty. It's a quantitative question that depends on the numerical values of the probabilities involved. If you want to prove a result, you'll have to add more assumptions - something beyond just knowing that the intersection is non-empty.
 

1. What is conditional probability?

Conditional probability is a mathematical concept that measures the likelihood of an event occurring given that another event has already occurred. It is a way to understand how one event can affect the probability of another event happening.

2. How is conditional probability calculated?

Conditional probability is calculated by dividing the probability of the joint occurrence of two events by the probability of the first event. This can be represented by the formula P(A|B) = P(A and B)/P(B), where P(A|B) is the conditional probability of event A given event B.

3. What is the difference between conditional probability and joint probability?

Conditional probability measures the likelihood of one event occurring given that another event has already occurred, while joint probability measures the likelihood of two events occurring simultaneously. Conditional probability is calculated using joint probability and the probability of the first event.

4. How can conditional probability be used in real life?

Conditional probability can be used in real life to make predictions and informed decisions. For example, it can be used in weather forecasting to predict the likelihood of rain given certain atmospheric conditions. It can also be used in medical diagnosis to determine the probability of a disease based on certain symptoms.

5. Can conditional probability be greater than 1?

No, conditional probability cannot be greater than 1. This is because it represents the likelihood of an event occurring, and the highest probability an event can have is 1 or 100%. If the conditional probability is greater than 1, it means the event is certain to occur and there is no uncertainty involved.

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