Probability Calculation for a String Cut in Half

The problem posed is to determine the probability that when a string of length 1 meter is cut at a random point, the longer piece of the string is at least twice the length of the shorter piece. This article explores the mathematical steps involved in solving this problem.

Defining the Problem

Let's define the problem contextually:

Total String Length: L 1 meter Random Cut: Let 'x' be the length of one piece after the cut, where x can vary from 0 to 1. Lengths of Pieces: The two resulting pieces will be 'x' and '1 - x'.

Identifying the Conditions

For the longer piece to be at least twice the length of the shorter piece, we need to consider two cases:

Case 1: x ≤ 0.5

In this case, '1 - x' is the longer piece. Therefore, the condition is:

Condition: 1 - x ≥ 2x Derivation: 1 ≥ 3x, thus x ≤ 1/3

Case 2: x > 0.5

Here, 'x' is the longer piece. The condition is:

Condition: x ≥ 2(1 - x) Derivation: x ≥ 2 - 2x, thus 3x ≥ 2, hence x ≥ 2/3

Determining the Valid Intervals

Based on the conditions derived, the valid intervals are:

Interval for Case 1: 0 to 1/3 Interval for Case 2: 2/3 to 1

Calculating the Lengths of Valid Intervals

Length of Interval for Case 1: 1/3 - 0 1/3 Length of Interval for Case 2: 1 - 2/3 1/3

Total Length of Valid Intervals

The total length of the valid intervals is:

1/3 1/3 2/3

Calculating the Probability

The probability that the longer piece is at least twice the length of the shorter piece is given by the ratio of the total length of the valid intervals to the total length of the string:

P (2/3) / 1 2/3

Therefore, the probability that the longer piece is at least twice the length of the shorter piece is 2/3.

Conclusion

This article has detailed the mathematical approach to solving the probability problem of a string cut at a random point, ensuring a deeper understanding of how probabilities can be calculated in such scenarios.

Related Keywords: string cut, probability, length distribution