(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 \(x = \cos 30°\)...
Question 1 Report
(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 : \(\frac{x^{2} + y^{2}}{y^{2} - x^{2}}\).
Answer Details
(a) Two dice give \(6\times6 = 36\) equally likely outcomes.
(i) Same digit: the pairs are (1,1),(2,2),...,(6,6), that is 6 outcomes.
\[P(\text{same digit}) = \tfrac{6}{36} = \tfrac{1}{6}.\]
(ii) Total greater than 5: count the totals that are 5 or less: total 2 (1 way), 3 (2), 4 (3), 5 (4), giving \(1+2+3+4 = 10\) outcomes.
\[P(\text{total} > 5) = \tfrac{36-10}{36} = \tfrac{26}{36} = \tfrac{13}{18}.\]
(b) \(x = \cos30^\circ = \tfrac{\sqrt3}{2}\Rightarrow x^2 = \tfrac{3}{4};\quad y = \sin30^\circ = \tfrac{1}{2}\Rightarrow y^2 = \tfrac{1}{4}.\)
\[\frac{x^2 + y^2}{y^2 - x^2} = \frac{\tfrac{3}{4}+\tfrac{1}{4}}{\tfrac{1}{4}-\tfrac{3}{4}} = \frac{1}{-\tfrac{1}{2}} = -2.\]
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