Conditional Probability: The "Given That" World
Master the art of solving "given that" probability problems with Bayes' Theorem and real-world applications for JEE success.
Why Conditional Probability Matters in JEE
Conditional probability appears in 2-3 questions per JEE paper, making it essential for scoring well. Understanding "given that" scenarios helps you:
- Solve complex real-world probability problems
- Apply Bayes' Theorem effectively
- Avoid common probability pitfalls
- Gain 4-6 marks with confidence
The Fundamental Formula
$P(A|B) = \frac{P(A \cap B)}{P(B)}$
Probability of A given B equals Probability of A and B divided by Probability of B
Understanding "Given That"
Conditional probability $P(A|B)$ means we're only considering the cases where B has already happened.
Key Insight:
The sample space shrinks to only those outcomes where B occurs.
Example: Dice Problem
When rolling a fair die, what is the probability of getting a 3 given that the number is odd?
Step 1: Identify events:
• A = Getting a 3
• B = Getting an odd number {1, 3, 5}
Step 2: Apply formula: $P(A|B) = \frac{P(A \cap B)}{P(B)}$
Step 3: Calculate:
• $P(A \cap B) = P(\text{3}) = \frac{1}{6}$
• $P(B) = P(\text{odd}) = \frac{3}{6} = \frac{1}{2}$
Step 4: $P(A|B) = \frac{1/6}{1/2} = \frac{1}{3}$
Independent vs Dependent Events
Understanding when events affect each other is crucial for correct probability calculations.
Independent Events:
$P(A|B) = P(A)$ - Knowing B occurred doesn't change A's probability
Dependent Events:
$P(A|B) \neq P(A)$ - B's occurrence affects A's probability
Example: Card Problem
From a standard deck, what's the probability of drawing a king given that the card is a face card?
Step 1: Identify events:
• A = Drawing a king
• B = Drawing a face card (Jack, Queen, King)
Step 2: $P(A|B) = \frac{P(A \cap B)}{P(B)} = \frac{P(\text{King})}{P(\text{Face Card})}$
Step 3: Calculate:
• Kings that are face cards: 4 cards
• Total face cards: 12 cards
Step 4: $P(A|B) = \frac{4}{12} = \frac{1}{3}$
Bayes' Theorem
The powerful theorem for reversing conditional probabilities.
Bayes' Theorem Formula
$P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)}$
When to Use Bayes' Theorem:
- Medical testing (disease given positive test)
- Quality control (defective given failed test)
- Spam filtering (spam given certain words)
Example: Medical Testing
A disease affects 1% of population. Test is 99% accurate (99% true positive, 99% true negative). If someone tests positive, what's the probability they actually have the disease?
Step 1: Define events:
• D = Has disease, $P(D) = 0.01$
• T+ = Tests positive
Step 2: Known probabilities:
• $P(T+|D) = 0.99$ (true positive rate)
• $P(T+|\text{not } D) = 0.01$ (false positive rate)
Step 3: Apply Bayes':
$P(D|T+) = \frac{P(T+|D)P(D)}{P(T+|D)P(D) + P(T+|\text{not } D)P(\text{not } D)}$
Step 4: Calculate:
$P(D|T+) = \frac{0.99 \times 0.01}{0.99 \times 0.01 + 0.01 \times 0.99} = \frac{0.0099}{0.0099 + 0.0099} = 0.5$
Surprising Result: Only 50% chance of actually having disease despite positive test!
🚀 Conditional Probability Problem-Solving Strategies
For "Given That" Problems:
- Always identify what's "given" first
- Shrink your sample space to the given condition
- Use Venn diagrams for visualization
- Check if events are independent
For Bayes' Theorem Problems:
- Clearly define all events
- Calculate total probability carefully
- Use tree diagrams for complex cases
- Always interpret your final answer
Concepts 4-5 Available in Full Version
Includes Total Probability Theorem and Advanced Bayes' Applications with JEE-level problems
📝 Quick Self-Test
Try these JEE-level conditional probability problems:
1. In a family with two children, given that at least one is a boy, what's the probability that both are boys?
2. A bag contains 3 red and 4 blue balls. Two balls are drawn without replacement. What's the probability the second is red given that the first was blue?
3. Three machines produce items. Machine A makes 50% with 2% defective, B makes 30% with 3% defective, C makes 20% with 4% defective. If an item is defective, what's the probability it came from machine A?
Ready to Master Conditional Probability?
Get complete access to all concepts with step-by-step video solutions and JEE practice problems