Conditional Probability Formula

[Avichal]

P(A/B) is defined to be P(A∩B)/P(B)

Why is this true?
When A and B are dependent events, I can understand why this is correct. It is clear when you see the venn diagram.
But for independent events, why is the formula correct? Any intuition or formal proof?

 

[mfb]

"Dependent events" always considers all cases, so independent events are a subset of those. If you understand the general case, using it in a special case should be no problem.
Anyway, it is a definition, asking about "correct or not" is meaningless.

 

[PeroK]

P(A/B) is defined to be P(A∩B)/P(B)

Why is this true?
When A and B are dependent events, I can understand why this is correct. It is clear when you see the venn diagram.
But for independent events, why is the formula correct? Any intuition or formal proof?

You can see this as follows:

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

Think about B happening "first".

If A and B both happen, then B must happen, then A must happen (given B has happened).

If A and B are independent, then $P(A/B) = P(A); \ P(A \cap B) = P(A)P(B)$ and the equation holds.

 

[chiro]

Hey Avichal.

The easiest way to convince yourself of it being true is to remember that if two events are independent, then one event happening will not in any way change the probability of another happening and vice-versa.

In mathematical notation this is defined as P(A|B) = P(A) given that B is a valid event (with a non-zero probability). If we use the definition of conditional probability along with this constraint we get:

P(A|B) = P(A and B)/P(B) = P(A) which implies
P(A and B) = P(A)*P(B) after multiplying both sides by P(B).

 

[blue_raver22]

The conditional probability formula, P(A/B) = P(A∩B)/P(B), is true for both dependent and independent events. This can be understood intuitively by looking at the definition of conditional probability. P(A/B) is the probability of event A occurring given that event B has already occurred. Therefore, it makes sense that this probability would be equal to the probability of both A and B occurring (P(A∩B)), divided by the probability of B occurring (P(B)). This is because the intersection of A and B represents the outcomes where both events occur, and we are only interested in the probability of A occurring within this subset of outcomes.

In the case of independent events, the formula still holds true. This can be seen through a formal proof using the definition of independence. If A and B are independent events, then P(A∩B) = P(A) * P(B). Substituting this into the formula, we get P(A/B) = (P(A) * P(B)) / P(B). The P(B) terms cancel out, leaving us with P(A/B) = P(A), which is the same as the probability of A occurring without any knowledge of B. This makes sense, as the occurrence of B does not affect the probability of A occurring if the events are independent.

In summary, the conditional probability formula is true for both dependent and independent events and can be intuitively understood and formally proven. It is a fundamental concept in probability theory and is used in a variety of applications, including in the fields of science and statistics.