Consider the following modified algorithm for binary search, which we will call $\textit{weighted binary search}$:

\begin{tabular}{l}
01: \textbf{FUNCTION} SEARCH($L$, value) \\
02:$\qquad$  hi $\leftarrow$ $\operatorname{len}(L) - 1$ \\
03:$\qquad$  lo $\leftarrow$ 0 \\
04:$\qquad$  \textbf{WHILE} hi $\geq$ lo \\
05:$\qquad\qquad$ guess $\leftarrow$ $\lfloor w \cdot\text{lo} + (1-w) \cdot \text{hi}\rfloor$ \\
06:$\qquad\qquad$ mid $\leftarrow$ $L[\text{guess}]$ \\
07:$\qquad\qquad$ \textbf{IF} mid $> \text{value}$ \\
08: $\qquad\qquad\qquad$ hi $\leftarrow$ $\text{guess} - 1$ \\
09: $\qquad\qquad$ \textbf{ELSE IF} mid $< \text{value}$ \\
10: $\qquad\qquad\qquad$ lo $\leftarrow$ $\text{guess} + 1$ \\
11: $\qquad\qquad$ \textbf{ELSE} \\
12: $\qquad\qquad\qquad$ \textbf{RETURN} guess \\
13:$\qquad$ \textbf{RETURN} -1 (not found) 
\end{tabular}\\

Assume $L$ is a list of the integers $\{1,2,\ldots,100\}$, in that order. Further assume that accessing the $k$th index of $L$ costs $k+1$ tokens (e.g. $L[0]$ costs $1$ token). Let $S$ be the set of all $w\in[\tfrac12,1)$ which minimize the average cost when $\texttt{value}$ is an integer selected at random in the range $[1,50]$. Given that $S=\left(x,\tfrac {74}{99}\right]$, determine $x$.
To solve this problem, we need to analyze the weighted binary search algorithm and determine the value of \( x \) such that the average cost is minimized for \( w \in [\frac{1}{2}, 1) \).

1. **Understanding the Weighted Binary Search Algorithm:**
   - The algorithm modifies the standard binary search by using a weighted average to determine the next guess index.
   - The guess index is calculated as \( \text{guess} = \lfloor w \cdot \text{lo} + (1-w) \cdot \text{hi} \rfloor \).

2. **Cost Analysis:**
   - Accessing the \( k \)-th index of \( L \) costs \( k+1 \) tokens.
   - We need to minimize the average cost when the value is an integer selected at random in the range \([1, 50]\).

3. **Range of \( w \):**
   - We are given that \( w \in [\frac{1}{2}, 1) \).
   - We need to find the set \( S \) of all \( w \) that minimize the average cost.

4. **Given Information:**
   - It is given that \( S = \left(x, \frac{74}{99}\right] \).
   - We need to determine \( x \).

5. **Analyzing the Cost Function:**
   - The cost function depends on the value of \( w \) and how it affects the guess index.
   - For \( w = \frac{1}{2} \), the algorithm behaves like a standard binary search.
   - As \( w \) increases towards 1, the guess index shifts towards the lower end of the current search range.

6. **Minimizing the Average Cost:**
   - To minimize the average cost, we need to balance the cost of accessing indices with the number of accesses.
   - The optimal \( w \) will minimize the total cost over all possible values in the range \([1, 50]\).

7. **Determining \( x \):**
   - Given that \( \max S = \frac{74}{99} \), we need to find the lower bound \( x \) of the interval.
   - The value \( \frac{74}{99} \) is approximately 0.7475.
   - We need to find the value of \( w \) just below \( \frac{74}{99} \) that still minimizes the average cost.

8. **Conclusion:**
   - Based on the given information and the analysis, the value of \( x \) is determined to be \( \frac{1}{2} \).

The final answer is \( \boxed{ x = \frac{1}{2} } \).