Problem: An ordered pair of integers, each of which has absolute value less than or equal to five, is chosen at random, with each such ordered pair having an equal likelihood of being chosen. What is the probability that the equation will not have distinct positive real roots?
Answer Choices:
A.
B.
C.
D.
E. none of these
Solution:
These six statements are equivalent:
the equation has positive roots;
the equation has real roots, the smaller of which is positive;
is real and ;
and ;
and ;
and ; or and or ; or and or or and or
.
The roots corresponding to the pairs ( ) described in (6) will be distinct unless . Thus, deleting and from the list in (6) yields the ten pairs resulting in distinct positive roots.
The desired probability is then .