SOLUTION TO LAST MONTH'S QUIZ
1). 1/2
2). 1/4
3). 1/8
Notes
Suppose x, y, z stand for the decimal parts of c, g, s
respectively. We may assume that each of x, y, z, is
distributed uniformly between 0 and 1.
1). x must lie between 0.25 and 0.75, so probability = 1/2.
2). The condition is satisfied if:
(i) neither c nor g is rounded up, but (c+g) is rounded
up, i.e. x < 0.5, y < 0.5, x+y > 0.5
or (ii) both c and g are rounded up, but (c+g) is not
doubly rounded up,
i.e. x > 0.5, y > 0.5, x+y < 1.5
The coloured areas on the diagram show the values of x and y
which satisfy these conditions. Total = 1/8 + 1/8 = 1/4.
3). The condition is satisfied only if c, g, s are all
rounded up, i.e. x > 0.5, y > 0.5, z > 0.5.
But x, y, x are not independent. In fact x+y+z must = 2.
So taking x and y as the independent variables, we
require: x > 0.5, y > 0.5, 2xy > 0.5.
This corresponds to the red area in the diagram,
whose area is 1/8.
