South 고등학교 3학년 Probability and Statistics
Chapters: 3
1. Counting
Circular and repeated permutations · Repeated combinations · Binomial theorem
- Permutations and Combinations – Counting without listing is the heart of this chapter. The fundamental principle of counting says: if one job can be done in m ways and the next in n ways, both together can be done in m × n ways. n! (n factorial) is 1 × 2 × … × n, with 0! = 1. A permutation is an arrangement, where order matters: the number of ways to arrange r things out of n different things is ⁿPᵣ = n!/(n − r)!. A combination is a selection, where order does not matter: ⁿCᵣ = n!/(r!(n − r)!). Each selection of r things can be arranged in r! ways, so ⁿPᵣ = ⁿCᵣ × r!. Useful facts: ⁿCᵣ = ⁿCₙ₋ᵣ and ⁿCᵣ + ⁿCᵣ₋₁ = ⁿ⁺¹Cᵣ. When some objects repeat, divide by the factorial of each repeat count.
- Binomial Theorem for Positive Integers – A binomial is a two-term expression like a + b. The binomial theorem tells us how to expand (a + b)ⁿ for any positive integer n without multiplying again and again: (a + b)ⁿ = ⁿC₀aⁿ + ⁿC₁aⁿ⁻¹b + ⁿC₂aⁿ⁻²b² + … + ⁿCₙbⁿ. There are n + 1 terms. The power of a goes down by 1 and the power of b goes up by 1 in each term, and the two powers always add up to n. The coefficients ⁿCᵣ are the numbers in row n of Pascal's triangle, where each number is the sum of the two above it. The (r + 1)th term is Tᵣ₊₁ = ⁿCᵣ aⁿ⁻ʳ bʳ. The pattern was known long ago in India (Pingala's Meru Prastara) and in Persia and China, and Pascal later studied the triangle in detail. The theorem is proved by mathematical induction using ⁿCᵣ₋₁ + ⁿCᵣ = ⁿ⁺¹Cᵣ.
2. Probability
Statistical vs mathematical probability · Basic properties · Addition rule · Complements · Conditional probability · Independence · Multiplication rule
- Probability: Events, Algebra of Events and Axioms – An event is any subset of the sample space S. From events A and B we build new events: not A (A′), A and B (A ∩ B), A or B (A ∪ B). Events are mutually exclusive if they share no outcome and exhaustive if together they cover S. The axiomatic approach says every P(E) ≥ 0, P(S) = 1, and for mutually exclusive A, B, P(A ∪ B) = P(A) + P(B). From these follow P(A′) = 1 − P(A) and P(A ∪ B) = P(A) + P(B) − P(A ∩ B).
- Conditional Probability, Multiplication Rule and Independent Events – Conditional probability is the chance of A when we already know B has happened. We throw away every outcome outside B and count again: P(A|B) = P(A ∩ B) ÷ P(B). Turned around, this gives the multiplication rule P(A ∩ B) = P(B)·P(A|B). If knowing B does not change the chance of A, the events are independent and P(A ∩ B) = P(A)·P(B).
3. Statistics
Random variables · Expected value and SD · Binomial distribution · Normal distribution · Population and sampling · Sample means · Estimating the mean
- Random Variables and Probability Distributions – A random variable X gives a number to every outcome of a chance experiment. A discrete random variable takes separate values; its probability distribution lists each value x with its probability P(X = x), and these add up to 1. The mean (expected value) is E(X) = Σ x·P(x). The variance Var(X) = E(X²) − [E(X)]² measures spread, and the standard deviation is its square root.
- Binomial Distribution – Repeat the same yes/no trial n times, independently, with the same chance of success p each time. The number of successes X follows the binomial distribution B(n, p): P(X = k) = ⁿCₖ pᵏ qⁿ⁻ᵏ with q = 1 − p. Its mean is np and its variance is npq.
- Normal Distribution – A normal distribution is a continuous, symmetric, bell-shaped distribution fixed by its mean μ (centre) and standard deviation σ (spread): X ~ N(μ, σ²). Mean = median = mode. About 68% of values lie within 1σ of μ, 95% within 2σ and 99.7% within 3σ. Any normal value is turned into a standard score z = (x − μ)/σ, which follows N(0, 1); probabilities are areas under the curve, read from a table or calculator as Φ(z).
- Sampling: Learning About a Population from a Sample – A population is the whole group we want to know about; a sample is a smaller part we actually check. A good sample is chosen at random so that it represents the population. Simple random sampling gives everyone an equal chance; stratified sampling takes the right share from each group; systematic sampling takes every k-th item. Different samples give slightly different answers (sampling variation), but bigger samples wobble less (law of large numbers). A biased sample gives a wrong answer however big it is.
- Sampling Distributions and the Central Limit Theorem – A statistic (like a sample mean x̄ or a sample proportion p̂) changes from sample to sample. If you took every possible sample and plotted the statistic, you would get its sampling distribution. Its centre is the true population value (μ or p), its spread is the standard error (σ/√n for means, √(p(1−p)/n) for proportions), and by the Central Limit Theorem its shape becomes close to normal when n is large enough.
- Confidence Intervals – A confidence interval is a range of believable values for an unknown population number (a mean μ or a proportion p), worked out from one sample. It has the shape estimate ± margin of error, where margin of error = critical value × standard error. A 95% level means the method catches the true value in about 95% of samples. Higher confidence gives a wider interval; a bigger sample gives a narrower one. If a claimed value lies outside the interval, the data give evidence against the claim.