Theory question 114
Probability(a) A bag contains 5 blue, 4 green and 3 yellow balls. All the balls are identical except for colour. Three balls are drawn at random without replacement. Find the probability that: (i) all three balls have the same colour; (ii) two balls have the same colour.
(b) The table shows the ranks of the marks scored by 7 candidates in Physics and Chemistry tests.
Physics: 6, 5, 4, 3, 2, 7, 1
Chemistry: 7, 6, 2, 4, 1, 5, 3
Calculate the Spearman's rank correlation coefficient.
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(a) ¹²C₃ = 220 selections.
(i) ⁵C₃ + ⁴C₃ + ³C₃ = 10 + 4 + 1 = 15; P = 15/220 = 3/44.
(ii) Exactly two the same: ⁵C₂ × 7 + ⁴C₂ × 8 + ³C₂ × 9 = 70 + 48 + 27 = 145; P = 145/220 = 29/44.
(b) d = −1, −1, 2, −1, 1, 2, −2; Σd² = 16.
ρ = 1 − 6(16)/(7(49 − 1)) = 1 − 96/336 = 0.714.
Spearman: ρ = 1 − 6Σd²/(n(n² − 1)).