(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. F...
Question 1 Report
(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.
Answer Details
(a) Bag: 5 blue, 4 green, 3 yellow (12 balls), draw 3 without replacement. Total selections \(={}^{12}C_3=220\).
(i) All three the same colour.\[{}^{5}C_3+{}^{4}C_3+{}^{3}C_3=10+4+1=15\]\[P=\frac{15}{220}=\frac{3}{44}\approx 0.068\]
(ii) Exactly two of the same colour (two alike and one different):\[{}^{5}C_2(7)+{}^{4}C_2(8)+{}^{3}C_2(9)=10(7)+6(8)+3(9)=70+48+27=145\]\[P=\frac{145}{220}=\frac{29}{44}\approx 0.659\](Check: all-different \(=5\times4\times3=60\); \(15+145+60=220\).)
(b) Spearman's rank correlation (Physics vs Chemistry ranks). \(n=7\).
| Physics | Chemistry | \(d\) | \(d^2\) |
|---|---|---|---|
| 6 | 7 | -1 | 1 |
| 5 | 6 | -1 | 1 |
| 4 | 2 | 2 | 4 |
| 3 | 4 | -1 | 1 |
| 2 | 1 | 1 | 1 |
| 7 | 5 | 2 | 4 |
| 1 | 3 | -2 | 4 |
| \(\sum d^2\) | 16 | ||
\[r_s=1-\frac{6\sum d^2}{n(n^2-1)}=1-\frac{6(16)}{7(48)}=1-\frac{96}{336}=1-0.286=0.714\]
Answers: (a)(i) \(\tfrac{3}{44}\); (a)(ii) \(\tfrac{29}{44}\); (b) \(r_s\approx 0.714\).
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