Conditional Probability (confusing)

In summary, the conversation is discussing the truth of the statement "P(A|B) + P(Ac|B) = 1" where [Ac] represents the complement of set A and B is assumed to have a probability greater than 0. They also mention two similar statements, the first being true and the second being not true. The third statement is also mentioned and it is stated that it is true. Steps for proving or disproving these statements are also discussed.
  • #1
sampahmel
21
0
Dear all,

P (A |B) + P (A c|B) = 1 [A c] denotes complement of set A and of course P (B)>0

Is the above statement true?


How about the following two:
P (A |B) + P (A |B c) = 1


P (C ∪ D|B) = P (C |B) + P (D|B) − P (C ∩ D|B)
 
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  • #2
What steps have you taken in proving/disproving these?
 
  • #3
first statement is true

P (A |B) + P (A |B c) = 1 is not true

if P(A) = x and A is independent of B, then P(A|B) = P(A|Bc) = x

i think the third statement is true
 
  • #4
TIP:
[tex]Pr(A|B)=\frac{Pr(A \cap B)}{Pr(B)} \quad Pr(B)>0[/tex]
[tex]Pr(A \cup B)=Pr(A)+Pr(B)-Pr(A \cap B)[/tex]
 

1. What is conditional probability?

Conditional probability is the likelihood of an event occurring given that another event has already occurred. It is calculated by dividing the probability of both events occurring by the probability of the initial event occurring.

2. How is conditional probability different from regular probability?

Regular probability is the likelihood of an event occurring without any prior knowledge or conditions. Conditional probability takes into account the occurrence of another event and adjusts the probability accordingly.

3. What is the formula for calculating conditional probability?

The formula for conditional probability is P(A|B) = P(A∩B) / P(B), where P(A|B) represents the probability of event A occurring given that event B has already occurred, P(A∩B) represents the probability of both events A and B occurring, and P(B) represents the probability of event B occurring.

4. What are some real-world applications of conditional probability?

Conditional probability is used in various fields, such as finance, medicine, and weather forecasting. It can be used to determine the likelihood of a patient having a certain disease based on their symptoms, or the chance of a stock price increasing given a certain market trend.

5. How can I improve my understanding of conditional probability?

One way to improve understanding of conditional probability is to practice solving problems and using the formula in different scenarios. Additionally, researching and learning about Bayesian probability and conditional independence can also deepen understanding of this concept.

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