1. What is Conditional Probability?

Conditional probability is the probability of an event occurring given that another event has already occurred. The notation P(A|B) represents the probability of event A occurring given that event B has occurred.

2. How Does the Calculator Work?

The calculator uses the conditional probability formula:

Where:

• \( P(A \mid B) \) — Probability of A given B
• \( P(A \cap B) \) — Probability of both A and B occurring
• \( P(B) \) — Probability of B occurring

Explanation: The formula calculates how the probability of A changes when we know that B has occurred.

3. Importance of Conditional Probability

Details: Conditional probability is fundamental in statistics, machine learning, and many real-world applications like medical testing, weather forecasting, and risk assessment.

4. Using the Calculator

Tips: Enter P(A ∩ B) and P(B) as values between 0 and 1. P(B) must be greater than 0 for the calculation to be valid.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between P(A|B) and P(B|A)?
A: P(A|B) is the probability of A given B, while P(B|A) is the probability of B given A. These are different unless P(A) = P(B).

Q2: Can conditional probability be greater than 1?
A: No, all probabilities must be between 0 and 1. If your calculation gives >1, check your inputs.

Q3: What if P(B) = 0?
A: The conditional probability is undefined when P(B) = 0 because you can't condition on an impossible event.

Q4: How is this related to Bayes' Theorem?
A: Bayes' Theorem is derived from the definition of conditional probability and allows us to "reverse" the conditioning.

Q5: What are some real-world applications?
A: Medical test interpretation (probability of disease given test result), spam filtering (probability email is spam given certain words), and many more.