Statistics 1: Probability and Distributions

Statistics 1: Probability and Distributions — CIE A Level Mathematics 9709

1. Probability Fundamentals

Basic rules:

• 0 ≤ P(A) ≤ 1 for any event A
• P(A') = 1 − P(A) (complement)
• P(A ∪ B) = P(A) + P(B) − P(A ∩ B) (addition rule)
• If A and B mutually exclusive: P(A ∪ B) = P(A) + P(B)

Conditional probability: P(A|B) = P(A ∩ B) / P(B)

Independent events: P(A ∩ B) = P(A) × P(B) ⟺ A and B are independent.

Tree diagrams: Multiply along branches (AND); add across branches (OR).

2. Permutations and Combinations

Permutation (order matters): Number of ways to arrange r from n: nPr = n!/(n−r)!

Combination (order doesn't matter): Number of ways to choose r from n: C(n,r) = n!/[r!(n−r)!]

Example: How many ways can 3 books be chosen from 8? C(8,3) = 56.

How many ways can they be arranged? P(8,3) = 8×7×6 = 336.

Arrangement with repetition: n! for n objects; n!/k! if k are identical; n!/(a!b!c!) if multiple groups of identical objects.

Example: How many arrangements of KARACHI? 7 letters, A appears twice.

7!/2! = 2520

3. Discrete Random Variables

A discrete random variable X takes specific values with given probabilities.

Probability distribution table: lists P(X = x) for all x.

Requirements:

• 0 ≤ P(X = x) ≤ 1 for all x
• ΣP(X = x) = 1

Expectation (mean): E(X) = Σ x·P(X = x)

Variance: Var(X) = E(X²) − [E(X)]² where E(X²) = Σ x²·P(X = x)

Standard deviation: σ = √Var(X)

Linear transformations:

• E(aX + b) = aE(X) + b
• Var(aX + b) = a²Var(X)

Example:

| X | 1 | 2 | 3 | 4 |

|---|---|---|---|---|

| P | 0.1 | 0.3 | 0.4 | 0.2 |

E(X) = 1(0.1) + 2(0.3) + 3(0.4) + 4(0.2) = 0.1 + 0.6 + 1.2 + 0.8 = 2.7

E(X²) = 1(0.1) + 4(0.3) + 9(0.4) + 16(0.2) = 0.1 + 1.2 + 3.6 + 3.2 = 8.1

Var(X) = 8.1 − (2.7)² = 8.1 − 7.29 = 0.81

4. The Binomial Distribution

If X counts the number of successes in n independent trials each with probability p of success:

X ~ B(n, p)

$$P(X = r) = inom{n}{r} p^r (1-p)^{n-r}$$

Mean: E(X) = np

Variance: Var(X) = np(1−p)

Standard deviation: σ = √[np(1−p)]

When to use Binomial:

1. Fixed number of trials (n)
2. Each trial is independent
3. Each trial has exactly two outcomes (success/failure)
4. Probability of success (p) is constant

Example: A fair die is thrown 10 times. Find P(exactly 3 sixes).

X ~ B(10, 1/6). P(X = 3) = C(10,3)(1/6)³(5/6)⁷

= 120 × (1/216) × (78125/279936) = 0.155 (3 s.f.)

Cumulative binomial: Use tables or calculator. P(X ≤ k) = Σᵣ₌₀ᵏ P(X = r).

P(X ≥ k) = 1 − P(X ≤ k−1)

Example: X ~ B(12, 0.3). Find P(X ≥ 4).

P(X ≥ 4) = 1 − P(X ≤ 3) = 1 − [P(0)+P(1)+P(2)+P(3)] ≈ 1 − 0.4925 = 0.5075