Statistics and Probability
Statistics and probability in NEB Class 12 covers correlation and regression, basic probability, conditional probability, and the binomial distribution. The exam tests correct interpretation of coefficients as much as computation.
Core rules
- Karl Pearson correlation coefficient: r = Σ(x - x̄)(y - ȳ) / √[Σ(x - x̄)² Σ(y - ȳ)²], with -1 ≤ r ≤ 1. It measures linear association and is unchanged by adding constants or by changing units, except that reversing a variable's sign reverses r.
- Regression line of y on x: y - ȳ = b_yx(x - x̄), where b_yx = r(s_y/s_x). The line of x on y uses b_xy = r(s_x/s_y); both pass through (x̄, ȳ).
- P(A ∪ B) = P(A) + P(B) - P(A ∩ B). For independent events, P(A ∩ B) = P(A)P(B).
- Conditional probability: P(A|B) = P(A ∩ B)/P(B), P(B) > 0.
- Binomial: P(X = r) = C(n,r)p^r q^(n-r), q = 1 - p; mean np, variance npq.
Worked example
A fair die is rolled three times. Find the probability of exactly two sixes. Each roll is an independent trial with p = 1/6, q = 5/6, n = 3. So P(X = 2) = C(3,2)(1/6)²(5/6)¹ = 3 × 1/36 × 5/6 = 15/216 = 5/72.
Traps
- Saying mutually exclusive events are independent; they are not, unless one has probability zero.
- Confusing P(A|B) with P(B|A).
- Using the regression line of x on y to predict y.
- Forgetting that r has the same sign as both regression slopes.
- Using the binomial formula when trials are not independent or n is not fixed.