To find the probability that at least one of the numbers \( \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}, \ldots \) is located between \( r \) and \( s \), we can use the concept of geometric series and probability. Let's break down the problem step by step:
First, note that the numbers in the sequence \( \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}, \ldots \) are in decreasing order and approach zero. Therefore, for a number to be located between \( r \) and \( s \), it must be less than or equal to \( s \) and greater than or equal to \( r \).
The first term, \( \frac{1}{2} \), is the largest term in the sequence, and it's also the most likely to be within the range \( r \) to \( s \). So, we'll focus on this term for our calculations.
Now, the probability that \( \frac{1}{2} \) is within the range \( r \) to \( s \) is the length of that range (which is \( s - r \)) divided by the total range of possible values (which is \( 1 - 0 = 1 \)).
So, the probability that \( \frac{1}{2} \) is within the range \( r \) to \( s \) is:
\[ P\left(\frac{1}{2} \text{ is within } r \text{ to } s\right) = \frac{s - r}{1} = s - r. \]
Since \( r \) and \( s \) are chosen independently at random, we can consider their joint probability.
The probability that \( \frac{1}{2} \) is NOT within the range \( r \) to \( s \) is the complement of the probability that it is within the range:
\[ P\left(\frac{1}{2} \text{ is NOT within } r \text{ to } s\right) = 1 - P\left(\frac{1}{2} \text{ is within } r \text{ to } s\right) = 1 - (s - r) = 1 - s + r. \]
Now, we want at least one of the numbers from the sequence to be within the range \( r \) to \( s \). This is equivalent to the event that \( \frac{1}{2} \) is within the range \( r \) to \( s \) OR \( \frac{1}{4} \) is within the range \( r \) to \( s \) OR \( \frac{1}{8} \) is within the range \( r \) to \( s \), and so on.
The probabilities that these events occur are independent since \( r \) and \( s \) are chosen independently. Therefore, we can use the principle of inclusion-exclusion to calculate the probability of at least one of these events occurring.
The probability of at least one of the numbers being within the range \( r \) to \( s \) is:
\[ P(\text{At least one of the numbers is within } r \text{ to } s) \]
\[ = P\left(\frac{1}{2} \text{ is within } r \text{ to } s\right) + P\left(\frac{1}{4} \text{ is within } r \text{ to } s\right) + P\left(\frac{1}{8} \text{ is within } r \text{ to } s\right) + \ldots \]
\[ = (s - r) + (s - r) + (s - r) + \ldots \]
\[ = n \cdot (s - r), \]
where \( n \) is the number of terms in the sequence that we are considering. In this case, \( n \) is infinite.
Since \( r \) and \( s \) are both between 0 and 1, and \( n \) is infinite, the probability \( P(\text{At least one of the numbers is within } r \text{ to } s) \) approaches infinity, which doesn't make sense in a probability context.
Therefore, it's important to note that when dealing with infinite sequences and probabilities, we need to be careful and consider the convergence of the series. In this case, the probability of at least one of the numbers being within the range \( r \) to \( s \) is not well-defined due to the infinite nature of the series.