Understanding Conditional Probability P(A|B)

The question asks for the conditional probability P(A|B) when the probability of event B, P(B), is equal to 1. Conditional probability helps us understand the likelihood of one event occurring given that another event has already occurred.

Formula for Conditional Probability

The standard formula for calculating the conditional probability of event A occurring given that event B has occurred is:

$$P(A|B) = \frac{P(A \cap B)}{P(B)}$$

Here, $P(A \cap B)$ represents the probability that both event A and event B occur.

Applying the Given Condition: P(B) = 1

We are given that the probability of event B is 1, i.e., $P(B) = 1$. Let's substitute this value into the conditional probability formula:

$$P(A|B) = \frac{P(A \cap B)}{1}$$

This simplifies to:

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

Now, we need to figure out what $P(A \cap B)$ equals when $P(B) = 1$.

Analyzing the Intersection when P(B) = 1

If $P(B) = 1$, it means that event B is a certain event. In probability theory, a certain event is an event that is sure to occur. The set of outcomes corresponding to a certain event is the entire sample space, denoted by S. So, B is equivalent to the sample space S.

The intersection of event A and event B ($A \cap B$) represents the outcomes where both A and B occur. Since B is the entire sample space (as $P(B)=1$), the outcomes where both A and B occur are simply the outcomes where A occurs, because B is guaranteed to include all possible outcomes. Therefore, the intersection of A and B is the same as event A itself:

$$A \cap B = A \cap S$$

Since any event A is a subset of the sample space S, the intersection of A and S is just A.

$$A \cap S = A$$

Thus, when $P(B) = 1$, the intersection $A \cap B$ is equal to A.

Substituting Back to Find P(A|B)

Since $A \cap B = A$ when $P(B) = 1$, it follows that $P(A \cap B) = P(A)$.

Now, we substitute this back into the simplified conditional probability formula we derived earlier:

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

Becomes:

$$P(A|B) = P(A)$$

So, the conditional probability of event A given event B, when P(B) = 1, is equal to the probability of event A.

Summary of the Result

When event B is a certain event with probability $P(B) = 1$, the conditional probability $P(A|B)$ is equal to $P(A)$. This makes intuitive sense because if B is guaranteed to happen, knowing that B occurred doesn't change the probability of A occurring; the sample space for A given B is effectively the same as the original sample space for A.

Comparing our result with the given options, we find that $P(A|B) = P(A)$.

Revision Table: Key Probability Concepts

Additional Information on Conditional Probability

Conditional probability is a fundamental concept in statistics and probability theory. It helps update our beliefs about the likelihood of an event based on new information (the occurrence of another event). The formula $P(A|B) = \frac{P(A \cap B)}{P(B)}$ is valid only when $P(B) > 0$. If $P(B) = 0$, the event B is impossible, and the conditional probability $P(A|B)$ is typically considered undefined.

In the specific case discussed where $P(B) = 1$, event B does not restrict the sample space in a meaningful way for event A, because B covers all possible outcomes. This is why the probability of A given B is simply the probability of A.

Understanding conditional probability is crucial for Bayes' theorem, which is used extensively in areas like machine learning, medical diagnosis, and risk assessment to revise probabilities as more data becomes available.