Final answer:
When a point is chosen at random on a line segment, the probability that the ratio of the shorter to the longer segment is less than a given value can be found by determining the range of positions that satisfy the condition and dividing the length of the range by the total length of the line segment.
Step-by-step explanation:
When a point is chosen at random on a line segment of length L, it means that any point along the line segment has an equal chance of being selected. To find the probability that the ratio of the shorter to the longer segment is less than a given value, we need to consider the possible positions of the chosen point. Let's assume that the chosen point divides the line segment into two segments: the shorter segment and the longer segment. The ratio of the shorter to the longer segment can be expressed as a fraction, where the numerator is the length of the shorter segment and the denominator is the length of the longer segment.
To find the probability, we need to determine the range of positions on the line segment that satisfy the given condition. For example, if we want to find the probability that the ratio is less than 1/2, we need to identify the range of positions where the length of the shorter segment is less than half the length of the longer segment. This range will depend on the value of L. Once we have determined the range, we can calculate the probability by dividing the length of the range by the total length of the line segment L.