Advanced Problem-Solving Techniques in Probability
Master conditional probability, Bayes' theorem, random variables, and expectation with JEE Advanced level problems.
Why Master These Probability Techniques?
Probability questions in JEE Advanced test conceptual clarity and application skills. These 6 techniques cover over 90% of probability problems in recent JEE Advanced papers:
- Conditional Probability - Foundation for complex problems
- Bayes' Theorem - Reverse probability calculations
- Random Variables - Discrete and continuous distributions
- Expectation & Variance - Statistical measures
- Probability Distributions - Binomial, Poisson, Normal
- Geometric Probability - Area-based problems
Conditional Probability & Multiplication Theorem
$P(A|B) = \frac{P(A \cap B)}{P(B)}$, provided $P(B) > 0$
Key Formulas:
• $P(A \cap B) = P(A) \cdot P(B|A) = P(B) \cdot P(A|B)$
• $P(A_1 \cap A_2 \cap \cdots \cap A_n) = P(A_1) \cdot P(A_2|A_1) \cdot P(A_3|A_1 \cap A_2) \cdots$
Example: JEE Advanced 2022
A bag contains 5 red and 3 blue balls. Two balls are drawn without replacement. Find the probability that both balls are red.
Step 1: Let A = first ball red, B = second ball red
Step 2: $P(A) = \frac{5}{8}$
Step 3: $P(B|A) = \frac{4}{7}$ (after drawing one red ball)
Step 4: $P(A \cap B) = P(A) \cdot P(B|A) = \frac{5}{8} \times \frac{4}{7} = \frac{5}{14}$
Bayes' Theorem & Total Probability
$P(A_i|B) = \frac{P(A_i) \cdot P(B|A_i)}{\sum_{j=1}^n P(A_j) \cdot P(B|A_j)}$
Law of Total Probability:
If $A_1, A_2, \ldots, A_n$ form a partition of sample space, then
$P(B) = \sum_{i=1}^n P(A_i) \cdot P(B|A_i)$
Example: Medical Testing
A disease affects 1% of population. Test is 95% accurate for sick people and 90% accurate for healthy people. If a person tests positive, what's the probability they actually have the disease?
Step 1: Define events: D = has disease, T+ = tests positive
Step 2: Given: $P(D) = 0.01$, $P(T+|D) = 0.95$, $P(T+|\overline{D}) = 0.10$
Step 3: $P(T+) = P(D)P(T+|D) + P(\overline{D})P(T+|\overline{D})$
$= 0.01 \times 0.95 + 0.99 \times 0.10 = 0.1085$
Step 4: $P(D|T+) = \frac{0.01 \times 0.95}{0.1085} \approx 0.0876$ (only 8.76%)
Random Variables & Probability Distributions
A random variable X is a function from sample space to real numbers.
Types of Random Variables:
Discrete: Finite or countably infinite possible values
Continuous: Uncountably infinite possible values
Example: Binomial Distribution
A fair coin is tossed 5 times. Find the probability of getting exactly 3 heads.
Step 1: This follows binomial distribution with n=5, p=0.5
Step 2: $P(X=3) = \binom{5}{3} (0.5)^3 (0.5)^2$
Step 3: $P(X=3) = 10 \times 0.125 \times 0.25 = 0.3125$
Example: Poisson Distribution
If calls arrive at a call center at average rate of 2 per minute, find probability of exactly 3 calls in 2 minutes.
Step 1: For 2 minutes, $\lambda = 2 \times 2 = 4$
Step 2: $P(X=3) = \frac{e^{-4} \cdot 4^3}{3!}$
Step 3: $P(X=3) = \frac{e^{-4} \cdot 64}{6} \approx 0.1954$
🚀 Advanced Problem-Solving Strategies
For Conditional Probability:
- Always define events clearly
- Use tree diagrams for multi-stage problems
- Check if events are independent
- Remember $P(A|B) \neq P(B|A)$ in general
For Random Variables:
- Verify $\sum P(X=x_i) = 1$ for discrete
- Check $\int_{-\infty}^{\infty} f(x)dx = 1$ for continuous
- Know when to use which distribution
- Practice expectation and variance calculations
Techniques 4-6 Available in Full Version
Includes Expectation & Variance, Geometric Probability, and Continuous Distributions with JEE Advanced problems
📝 Quick Self-Test
Try these JEE-level problems to test your understanding:
1. Three cards are drawn without replacement from 52 cards. Find probability that all are aces.
2. If $P(A) = 0.4$, $P(B) = 0.5$, and $P(A \cup B) = 0.7$, are A and B independent?
3. A random variable X has probability distribution: $P(X=k) = c \cdot 2^{-k}$ for $k=1,2,3,\ldots$. Find c.
📚 Essential Probability Formulas
• $P(A \cup B) = P(A) + P(B) - P(A \cap B)$
• $P(A') = 1 - P(A)$
• $P(A \cap B) = P(A) \cdot P(B)$ if independent
• $E[X] = \sum x_i P(X=x_i)$
• $Var(X) = E[X^2] - (E[X])^2$
• Binomial: $P(X=k) = \binom{n}{k} p^k (1-p)^{n-k}$
Ready to Master All 6 Techniques?
Get complete access to all techniques with step-by-step video solutions and JEE Advanced practice problems