1. What is Conditional Probability?

Conditional probability is the probability of an event occurring given that another event has already occurred. It's a fundamental concept in probability theory and statistics that helps us understand the relationship between events.

2. How Does the Calculator Work?

The calculator uses the conditional probability formula:

Where:

• \( P(A|B) \) — Probability of event A occurring given that B is true
• \( P(A \cap B) \) — Probability of both A and B occurring
• \( P(B) \) — Probability of event 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 essential in many fields including medicine (diagnostic tests), machine learning (Bayesian networks), finance (risk assessment), and everyday decision making.

4. Using the Calculator

Tips: Enter valid probabilities between 0 and 1. P(A∩B) cannot be greater than P(B). Both values must be positive, and P(B) must be greater than 0.

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, probabilities always range between 0 and 1. If your calculation gives >1, check that P(A∩B) ≤ P(B).

Q3: What if P(B) = 0?
A: 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 real-world applications?
A: Weather forecasting, medical testing, spam filtering, and many machine learning algorithms use conditional probability.