(a) Two fair die are thrown once. Find the probabitlity of getting : (i) the same digit ; (ii) a total score greater than 5.
(b) Given that (x = cos 30°) and (y = sin 30°), evaluate without using a mathematical table or calculator : (frac{x^{2} + y^{2}}{y^{2} – x^{2}}).
Explanation
(a) Sample – space
| 1 | 2 | 3 | 4 | 5 | 6 | |
| 1 | 1, 1 | 1, 2 | 1, 3 | 1, 4 | 1, 5 | 1, 6 |
| 2 | 2, 1 | 2, 2 | 2, 3 | 2, 4 | 2, 5 | 2, 6 |
| 3 | 3, 1 | 3, 2 | 3, 3 | 3, 4 | 3, 5 | 3, 6 |
| 4 | 4, 1 | 4, 2 | 4, 3 | 4, 4 | 4, 5 | 4, 6 |
| 5 | 5, 1 | 5, 2 | 5, 3 | 5, 4 | 5, 5 | 5, 6 |
| 6 | 6, 1 | 6, 2 | 6, 3 | 6, 4 | 6, 5 | 6, 6 |
(i) Same digit = {(1,1), (2,2),(3,3), (4,4),(5,5), (6,6)}
P(same digit) = (frac{6}{36})
(ii) P(total score greater than 5) = (frac{26}{36})
= (frac{13}{18})
(b) (frac{x^{2} + y^{2}}{y^{2} – x^{2}} = frac{(cos 30)^{2} + (sin 30)^{2}}{(sin 30)^{2} – (cos 30)^{2}})
= (frac{(frac{sqrt{3}}{2})^{2} + (frac{1}{2})^{2}}{(frac{1}{2})^{2} – (frac{sqrt{3}}{2})^{2}})
= (frac{frac{3}{4} + frac{1}{4}}{frac{1}{4} – frac{3}{4}})
= (frac{1}{-frac{1}{2}})
= (-2).