Conditional Probability
Master the art of restricted outcomes and sequential events
Chapter Overview
Conditional probability is the probability of something happening given that something else has already happened. The key insight is that you must calculate probabilities from a smaller, restricted set of outcomes, not from all possible events.
- Basic Concept: Conditional probability restricts your sample space to only the outcomes where the given condition is true
- Notation: P(A|B) = "the probability of A given that B has already occurred"
- Combined Probabilities: When events happen one after another, you must adjust the number of remaining outcomes as you go
- Without Replacement: When items aren't put back (like drawing beads from a bag), the denominator changes after each draw
- The "Or" Rule: When multiple different sequences lead to your desired outcome, add their individual probabilities together
- Order Matters: AB (first A, then B) is different from BA (first B, then A)
1. What is Conditional Probability?
The Core Idea
A conditional probability is the probability of event A occurring given that event B has already happened. It's not about all possible outcomes — it's about outcomes in a restricted universe where B is already true.
Think of it this way: Imagine you're told "a number was selected from {2, 4, 6, 8}". The probability it's a multiple of 3 is now much smaller than if you had ALL numbers. You're working within a restricted set — that's conditional probability.
The Key Difference: Restricted vs. All Outcomes
Without restriction: If you pick a digit from {1, 2, 3, 4, 5, 6, 7, 8, 9}, the probability it's a multiple of 3 is 3/9 (because 3, 6, 9 work).
With restriction: If you pick a digit from {2, 4, 6, 8} (only even numbers), the probability it's a multiple of 3 is 1/4 (because only 6 works out of 4 possibilities).
Formal Notation
The notation P(A|B) means "the probability of A occurring given that B has already occurred". You don't need to memorise this for the exam, but it helps you understand what a question is asking.
A bag contains the numbers: {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. A number is picked at random. Find the probability that the number is even, given that it is greater than 5.
2. How to Calculate Conditional Probabilities
The Process
To find a conditional probability, follow these steps:
- Identify the condition. What has already happened? This defines your restricted set.
- Count the outcomes in the restricted set. This is your denominator.
- Count how many of those satisfy your target event. This is your numerator.
- Write the fraction. Target outcomes ÷ Restricted outcomes.
Worked Example: Venn Diagram
Problem: Sets A and B are shown below.
A = {2, 6, 12, 14, 28}
B = {7, 14, 21, 28, 35}
ξ = {1, 2, 5, 6, 7, 12, 14, 21, 28, 35}
A number is picked at random. Find the probability that the number is in set A, given that it is in set B.
Solution:
Step 1:
Step 2:
Step 3: Write the fraction: 2/5
Answer: 2/5
Common Mistake: