Deriving and Understanding the Conditional Probability Formula: P(A|B) P(A ∩ B)/P(B)
The formula for conditional probability, ( P(A|B) frac{P(A cap B)}{P(B)} ), is a fundamental concept in probability theory. This article will explore the proof of this formula using the definitions of probability and the concept of sample space.
Understanding Events
Let (A) and (B) be two events within a sample space (S). Here, (A) represents outcomes in event (A), and (B) represents outcomes in event (B).
The event (A cap B) represents the outcomes that are in both (A) and (B). This intersection indicates the outcomes that satisfy both events (A) and (B).
Definition of Conditional Probability
The conditional probability (P(A|B)) is defined as the probability of event (A) occurring under the condition that event (B) has occurred. This can be intuitively understood as restricting our attention to the outcomes in (B) and then looking for the outcomes that are also in (A).
Probability of Event B
The probability (P(B)) gives the measure of the likelihood of event (B) occurring. This is the total probability assigned to event (B).
Restricting the Sample Space
When we condition on (B), we effectively limit our sample space to the outcomes in (B). The total probability of (B) is (P(B)).
Calculating P(A|B)
The probability of event (A) given event (B) can be understood as the ratio of the probability of both (A) and (B) occurring compared to the probability of (B):
(P(A|B) frac{P(A cap B)}{P(B)})
This can be expressed mathematically as:
(P(A|B) frac{text{Probability of outcomes in both } Atext{ and } B}{text{Probability of outcomes in } B})
Derivation of the Formula
Starting with the general formula for conditional probability, we have:
(P(A|B) frac{P(A cap B)}{P(B)}), where (P(B) eq 0)
Now, let's move (P(B)) to the other side of the equation:
(P(A cap B) P(B) cdot P(A|B))
This can be interpreted as the probability of the simultaneous occurrence of events (A) and (B) being equal to the probability that (B) has occurred multiplied by the probability that (A) will occur given that (B) has occurred.
Conclusion
We have derived the formula for conditional probability, which is:
(P(A|B) frac{P(A cap B)}{P(B)}), where (P(B) eq 0)
This proof illustrates how conditional probability is fundamentally about focusing on the subset of outcomes defined by event (B) and determining how many of those outcomes also satisfy event (A).
Note: The probability formula for conditional probability assumes that (P(B) eq 0). If (P(B) 0), the formula is undefined because it involves division by zero.