Probability
1. Introduction
Probability quantifies uncertainty. This chapter covers advanced probability concepts including conditional probability, Bayes' theorem, random variables, and the binomial distribution — all essential for ISC mathematics.
2. Conditional Probability
P(A|B) = P(A ∩ B)/P(B), provided P(B) > 0.
It represents the probability of A given that B has already occurred.
3. Multiplication Theorem
P(A ∩ B) = P(A)·P(B|A) = P(B)·P(A|B) For independent events: P(A ∩ B) = P(A)·P(B)
4. Bayes' Theorem
P(Eᵢ|A) = P(Eᵢ)·P(A|Eᵢ) / Σ P(Eⱼ)·P(A|Eⱼ)
Bayes' theorem allows us to update probabilities based on new evidence.
'Bayes' theorem is used when we know the probabilities of causes and want to find the probability of a cause given an effect.'
5. Probability Distribution
5.1 Random Variable
A random variable X assigns a real number to each outcome of a random experiment. It can be discrete or continuous.
5.2 Probability Distribution
For a discrete random variable X taking values x₁, x₂, ..., xₙ with probabilities p₁, p₂, ..., pₙ:
Σ pᵢ = 1, pᵢ ≥ 0 for all i.
5.3 Mean and Variance
Mean (Expected value): μ = E(X) = Σ xᵢpᵢ Variance: σ² = E(X²) - [E(X)]² = Σ xᵢ²pᵢ - μ²
6. Bernoulli Trials
A Bernoulli trial has exactly two outcomes: success (p) and failure (q = 1 - p). Multiple independent Bernoulli trials with constant p give the Binomial distribution.
7. Binomial Distribution
If X ~ B(n, p), then P(X = r) = ⁿCᵣ pʳ q^{n-r}, r = 0, 1, 2, ..., n.
Mean = np, Variance = npq, Standard deviation = √(npq).
'For a binomial distribution, identify: n (number of trials), p (probability of success), and q = 1-p (probability of failure).'
8. Worked Problems
Problem 1: Bag A has 3 red, 5 white balls. Bag B has 4 red, 6 white balls. A bag is chosen at random and a ball is drawn which is red. Find the probability it came from bag A. Solution: P(A) = P(B) = 1/2. P(R|A) = 3/8, P(R|B) = 4/10 = 2/5. P(A|R) = (1/2 × 3/8)/(1/2 × 3/8 + 1/2 × 2/5) = (3/16)/(3/16 + 1/5) = (3/16)/((15+16)/80) = (3/16)/(31/80) = 15/31.
Problem 2: A random variable X has distribution: P(X=0)=0.1, P(X=1)=0.3, P(X=2)=0.4, P(X=3)=0.2. Find E(X) and Var(X). Solution: E(X) = 0×0.1 + 1×0.3 + 2×0.4 + 3×0.2 = 0 + 0.3 + 0.8 + 0.6 = 1.7. E(X²) = 0×0.1 + 1×0.3 + 4×0.4 + 9×0.2 = 0 + 0.3 + 1.6 + 1.8 = 3.7. Var(X) = 3.7 - (1.7)² = 3.7 - 2.89 = 0.81.
Problem 3: A die is rolled 5 times. Find the probability of getting exactly two sixes. Solution: n = 5, p = 1/6, q = 5/6. P(X = 2) = ⁵C₂(1/6)²(5/6)³ = 10 × (1/36) × (125/216) = 1250/7776 = 625/3888.
Problem 4: In an examination, 80% students pass. Find probability that in a group of 6 students, at most 2 fail. Solution: Let X = number of failures. p = 0.2, n = 6. P(X ≤ 2) = P(0) + P(1) + P(2) = ⁶C₀(0.2)⁰(0.8)⁶ + ⁶C₁(0.2)¹(0.8)⁵ + ⁶C₂(0.2)²(0.8)⁴ = 0.2621 + 0.3932 + 0.2458 = 0.9011.
9. Common Mistakes
'Students often forget that the sum of all probabilities in a distribution must equal 1. Use this to find missing probabilities.'
'When using Bayes' theorem, clearly identify all events and compute P(Eᵢ) and P(A|Eᵢ) separately before applying the formula.'
10. ISC Exam Focus
| Topic | Theory Marks | Practical Marks |
|---|---|---|
| Conditional probability | 3 | 2 |
| Bayes' theorem | 4 | 2 |
| Random variable and distribution | 4 | 3 |
| Binomial distribution | 4 | 3 |
11. Self-Test Questions
- Bag I has 2 white, 3 red balls. Bag II has 4 white, 2 red balls. A bag is chosen and two balls are drawn, both white. Find probability they came from Bag I.
- The mean and variance of a binomial distribution are 4 and 3 respectively. Find P(X = 0).
- The probability of hitting a target is 1/3. Find the probability that in 6 shots, the target is hit at most 2 times.
- 5 cards are drawn from a well-shuffled pack of 52 cards. Find the probability distribution of the number of aces.
- A pair of dice is thrown 4 times. Getting a sum of 9 is considered a success. Find the mean and variance of successes.
12. Quick Reference — Key Formulas
| Concept | Formula |
|---|---|
| Conditional probability | P(A |
| Multiplication theorem | P(A∩B) = P(A)·P(B |
| Bayes' theorem | P(Eᵢ |
| Binomial PMF | P(X=r) = ⁿCᵣ pʳ q^{n-r} |
| Binomial mean | μ = np |
| Binomial variance | σ² = npq |
| Expected value | E(X) = Σ xᵢpᵢ |
| Variance formula | Var(X) = E(X²) - [E(X)]² |
13. ISC Examination Strategy
- For Bayes' theorem problems, first identify the partition events (causes) and the effect event clearly.
- Always verify that Σ pᵢ = 1 for any probability distribution.
- In binomial problems, confirm that trials are independent and the probability of success remains constant.
- For conditional probability, check whether events are independent before multiplying.
- Practice problems involving "at least one", "at most", and "exactly" variations.
