Conditional Probability
Conditional probability measures the likelihood of an event occurring given that another event has already happened. This prior context shrinks the active sample space.
Fundamental Principles
Conditional Formula P(A|B)
The probability of event A occurring given that event B has already occurred is written as: P(A|B) = P(A ∩ B) / P(B), where P(B) must be greater than 0.
Multiplication Rule for Dependent Events
Calculated as: P(A ∩ B) = P(B) × P(A|B).
Essential Formulation Tips
- The most reliable way to solve conditional problems is to use the known condition to trim down your starting sample space denominator directly.
Shortcut Execution Techniques
- If two events are completely independent, then P(A|B) is simply equal to P(A), because the occurrence of event B has no effect on event A.
Contextual Inquiries (FAQs)
Q: What is the difference between P(A|B) and P(B|A)?
A:
Example Breakdown: Evaluating Sequential Pulls Without Replacement
Classic dependent probability problem frequently seen on advanced screening tests.Step 1: Note that the first red chip is already gone, changing the remaining pool size.
Step 2: Update the pool metrics: 4 red chips and 4 black chips are left, making 8 total chips.
Step 3: Calculate the probability of picking a black chip from the updated pool: 4 / 8 = 1 / 2 = 0.50.
Conditional Probability Practice Set 1
10 questions evaluating dependent events, shrinking sample spaces, and real-world conditional scenarios.
Q1. If P(A) = 0.6, P(B) = 0.4, and P(A ∩ B) = 0.2, calculate the value of P(A|B).
Q2. A card is drawn from a 52-card pack. Given that the drawn card is red, what is the probability that it is a King?
Q3. Two dice are rolled. Given that the sum of the faces is 8, find the probability that one of the dice shows a 3.
Q4. A family has two children. Given that at least one child is a boy, find the probability that both children are boys.
Q5. A drawer contains 6 black socks and 4 blue socks. Two socks are pulled out one after another without replacement. Find the probability that both are black.
Q6. If P(A) = 0.5 and P(B|A) = 0.3, find the value of the joint probability P(A ∩ B).
Q7. In a corporate team, 40% of members pass an AI test and 30% pass a cybersecurity test. If 15% pass both tests, find the probability that a member who passed the AI test also passed the cybersecurity test.
Q8. A student rolls a die. Given that the face showing is an even number, what is the probability that it is a 6?
Q9. A bag contains 3 red and 7 green balls. Two balls are drawn without replacement. Find the probability that the second ball is green, given that the first ball drawn was green.
Q10. Events X and Y are independent. If P(X) = 0.7 and P(Y) = 0.2, find the conditional probability P(X|Y).