If any of the pieces has length exactly 5 feet, the triangle will be completely flat. But the probability that a piece will be exactly 5 feet (or any particular length) is 0, so we can safely rule out that possibility. Also, if any length is greater than 5 feet, the triangle inequality will fail. Therefore, all three pieces must have length less than 5 feet. So we've reduced the question to finding the probability that all three pieces will be less than 5 feet in length.
The answer depends on where the saw cuts fall. The two cuts must fall on opposite sides of the midpoint of the pole, else one piece would be longer than 5 feet. Second, the distance between the two cuts must be less than 5 feet (or else the middle piece will be longer than 5 feet).
There are four possible ways to position the cuts on the left half or
right half of the pole. Two of the combinations have a cut on either
half. Thus the probability that the cuts lie on opposite halves of the
rod is 0.5. The distance between the cuts is either smaller than 5, or
bigger than 5 (eliminating the possibility of the distance being
exactly 5). These events are equally likely. Thus the probabilities
that the cuts are no more than 5 feet apart is 0.5. Therefore the
probability of both events happening is 0.25.