Theory question 105
Probability(a) A number is selected at random from each of the sets {2,3,4} and {1,3,5}. What is the probability that the sum of the two numbers will be less than 7 but greater than 3?
(b) In the diagram above, ABCD is a circle. DAE, CBE, ABF and DCF are straight lines. If y + m = 90°, find the value of x.
Show the model answer
Model answer
(a) The 9 equally likely sums are 3, 5, 7, 4, 6, 8, 5, 7, 9. Those greater than 3 and less than 7 are 5, 4, 6, 5: probability = 4/9.
(b) Let m = ∠AEB and x = ∠ABE = ∠CBF (vertically opposite).
Exterior angle of triangle ABE: ∠DAB = x + m. Exterior angle of triangle BCF: ∠DCB = x + y.
ABCD is cyclic, so (x + m) + (x + y) = 180°: 2x + 90° = 180°, x = 45°.
Explanation
Use exterior angles of the two small triangles, then opposite angles of the cyclic quadrilateral.