## Line Chart: Mean Pass Rate vs. Mean Number of Tokens Generated ### Overview The image is a line chart showing the relationship between the mean pass rate and the mean number of tokens generated. The chart includes data series for different values of 'np' (represented by lines) and 'nfr' (represented by markers). Error bars are present on the data points. A shaded region indicates uncertainty. ### Components/Axes * **X-axis:** "Mean number of tokens generated". The scale ranges from 0 to 10000, with tick marks at intervals of 2000. * **Y-axis:** "Mean pass rate". The scale ranges from 0.0 to 1.0, with tick marks at intervals of 0.2. * **Legend (Top-Left):** * Brown line: `np = 1` * Yellow line: `np = 2` * Teal line: `np = 5` * Cyan line: `np = 10` * Dark Blue line: `np = 25` * **Legend (Top-Right):** * Gray circle: `nfr = 1` * Gray inverted triangle: `nfr = 3` * Gray square: `nfr = 5` * Gray triangle: `nfr = 10` ### Detailed Analysis * **General Trend:** The mean pass rate generally increases as the mean number of tokens generated increases. The rate of increase appears to diminish as the number of tokens generated gets larger. * **Data Series Analysis:** * **nfr = 1 (Gray Circle):** Starts at approximately (0, 0.04), increases to approximately (6000, 0.15), and ends around (8000, 0.14). * **nfr = 3 (Gray Inverted Triangle):** Starts at approximately (1500, 0.07), increases to approximately (6000, 0.15), and ends around (8000, 0.15). * **nfr = 5 (Gray Square):** Starts at approximately (2500, 0.08), increases to approximately (6000, 0.14), and ends around (7000, 0.15). * **nfr = 10 (Gray Triangle):** Starts at approximately (3500, 0.10), increases to approximately (6000, 0.14), and ends around (8000, 0.14). * **Error Bars:** Error bars are present on each data point, indicating the uncertainty in the mean pass rate. The error bars appear to be relatively consistent in size across the different data series. * **Shaded Region:** A gray shaded region surrounds the data points, likely representing a confidence interval or standard deviation. ### Key Observations * The mean pass rate increases with the mean number of tokens generated, but the rate of increase slows down as the number of tokens increases. * The different values of 'nfr' do not appear to have a significant impact on the mean pass rate. The data points for different 'nfr' values are clustered closely together. * The error bars and shaded region indicate a degree of uncertainty in the mean pass rate. ### Interpretation The chart suggests that increasing the number of tokens generated generally leads to a higher pass rate. However, there appears to be a point of diminishing returns, where generating more tokens does not significantly improve the pass rate. The values of 'nfr' do not seem to have a strong influence on the pass rate, suggesting that other factors may be more important in determining the outcome. The uncertainty in the data, as indicated by the error bars and shaded region, should be considered when interpreting the results.

## Chart: Mean Pass Rate vs. Mean Number of Tokens Generated ### Overview This image displays a 2D scatter plot with error bars, overlaid on a dark gray line with a shaded confidence interval. The chart illustrates the relationship between "Mean number of tokens generated" on the x-axis and "Mean pass rate" on the y-axis. Individual data points are differentiated by color, representing `n_p` values, and shape, representing `n_fr` values, as defined in a dual-column legend. ### Components/Axes **X-axis:** * **Title:** "Mean number of tokens generated" * **Scale:** Ranges from 0 to 10000. * **Major Ticks:** 0, 2000, 4000, 6000, 8000, 10000. * **Labels:** Rotated approximately 45 degrees counter-clockwise for readability. **Y-axis:** * **Title:** "Mean pass rate" * **Scale:** Ranges from 0.0 to 1.0. * **Major Ticks:** 0.0, 0.2, 0.4, 0.6, 0.8, 1.0. **Grid:** * A light gray grid is present, extending from both major X and Y axis ticks across the plotting area. **Legend (Top-right corner):** The legend is divided into two conceptual columns, indicating how `n_p` values are mapped to colors (represented by line segments) and `n_fr` values are mapped to shapes. The legend is positioned within the upper-right quadrant of the plot area. * **Left Column (Colors for `n_p`):** * Brown line segment: `n_p = 1` * Goldenrod/Orange-yellow line segment: `n_p = 2` * Teal/Cyan line segment: `n_p = 5` * Bright Blue line segment: `n_p = 10` * Dark Blue line segment: `n_p = 25` (Note: No data points with this color are visible on the plot.) * **Right Column (Shapes for `n_fr`):** * Dark Gray Circle: `n_fr = 1` * Dark Gray Inverted Triangle: `n_fr = 3` * Dark Gray Square: `n_fr = 5` * Dark Gray Triangle: `n_fr = 10` **Overall Trend Line and Confidence Interval:** * A prominent dark gray line traverses the plot, representing an overall trend. * A semi-transparent gray shaded area surrounds this dark gray line, indicating a confidence interval or standard deviation for the trend. This line and shaded area are not explicitly labeled in the legend. ### Detailed Analysis The chart displays 11 distinct data points, each represented by a specific color (corresponding to `n_p`) and shape (corresponding to `n_fr`). Each point includes horizontal and vertical error bars, indicating uncertainty in both the "Mean number of tokens generated" and "Mean pass rate" respectively. The overall trend, represented by the dark gray line, shows an initial rapid increase in "Mean pass rate" as "Mean number of tokens generated" increases, followed by a flattening or slight increase at higher token counts. Here are the approximate values for each data point, identified by their `n_p` (color) and `n_fr` (shape) values: 1. **`n_p = 1` (Brown), `n_fr = 1` (Circle):** * Mean number of tokens generated: ~550 (horizontal error bar from ~450 to ~650) * Mean pass rate: ~0.04 (vertical error bar from ~0.03 to ~0.05) * This point is located in the lower-left quadrant, close to the origin. 2. **`n_p = 2` (Goldenrod), `n_fr = 3` (Inverted Triangle):** * Mean number of tokens generated: ~1250 (horizontal error bar from ~1150 to ~1350) * Mean pass rate: ~0.07 (vertical error bar from ~0.06 to ~0.08) * This point is to the right and slightly above the first point. 3. **`n_p = 1` (Brown), `n_fr = 5` (Square):** * Mean number of tokens generated: ~1850 (horizontal error bar from ~1750 to ~1950) * Mean pass rate: ~0.08 (vertical error bar from ~0.07 to ~0.09) * This point is further right and slightly above the previous point. 4. **`n_p = 2` (Goldenrod), `n_fr = 10` (Triangle):** * Mean number of tokens generated: ~2550 (horizontal error bar from ~2450 to ~2650) * Mean pass rate: ~0.10 (vertical error bar from ~0.09 to ~0.11) * This point continues the upward trend. 5. **`n_p = 5` (Teal), `n_fr = 1` (Circle):** * Mean number of tokens generated: ~3050 (horizontal error bar from ~2950 to ~3150) * Mean pass rate: ~0.11 (vertical error bar from ~0.10 to ~0.12) * This point is slightly above and to the right of the previous point. 6. **`n_p = 1` (Brown), `n_fr = 3` (Inverted Triangle):** * Mean number of tokens generated: ~3550 (horizontal error bar from ~3450 to ~3650) * Mean pass rate: ~0.10 (vertical error bar from ~0.09 to ~0.11) * This point shows a slight dip in pass rate compared to the previous point, despite more tokens. 7. **`n_p = 2` (Goldenrod), `n_fr = 5` (Square):** * Mean number of tokens generated: ~4050 (horizontal error bar from ~3950 to ~4150) * Mean pass rate: ~0.11 (vertical error bar from ~0.10 to ~0.12) * This point is slightly above the previous point. 8. **`n_p = 5` (Teal), `n_fr = 3` (Inverted Triangle):** * Mean number of tokens generated: ~4550 (horizontal error bar from ~4450 to ~4650) * Mean pass rate: ~0.12 (vertical error bar from ~0.11 to ~0.13) * This point continues the general upward trend. 9. **`n_p = 10` (Bright Blue), `n_fr = 1` (Circle):** * Mean number of tokens generated: ~6250 (horizontal error bar from ~6150 to ~6350) * Mean pass rate: ~0.16 (vertical error bar from ~0.15 to ~0.17) * This point represents a significant jump in tokens generated and a higher pass rate, aligning with the flattening part of the overall trend line. 10. **`n_p = 5` (Teal), `n_fr = 10` (Triangle):** * Mean number of tokens generated: ~6850 (horizontal error bar from ~6750 to ~6950) * Mean pass rate: ~0.16 (vertical error bar from ~0.15 to ~0.17) * This point is very close in pass rate to the previous point but with more tokens generated. 11. **`n_p = 2` (Goldenrod), `n_fr = 10` (Triangle):** * Mean number of tokens generated: ~8350 (horizontal error bar from ~8250 to ~8450) * Mean pass rate: ~0.13 (vertical error bar from ~0.12 to ~0.14) * This point shows a decrease in pass rate compared to the two preceding points, despite a higher number of tokens generated. ### Key Observations * **Overall Trend:** The "Mean pass rate" generally increases with the "Mean number of tokens generated," but the rate of increase diminishes significantly after approximately 4000-5000 tokens, with the curve flattening out. The maximum observed mean pass rate is around 0.16. * **Parameter Influence:** The scatter points suggest that both `n_p` (color) and `n_fr` (shape) influence the mean pass rate and the mean number of tokens generated. Higher `n_p` values (e.g., `n_p=10` in bright blue) tend to be associated with higher token counts and pass rates, while lower `n_p` values (e.g., `n_p=1` in brown) are generally at lower token counts and pass rates. * **`n_fr` Variation:** The shapes representing `n_fr` values are distributed across the range of tokens generated, indicating that `n_fr` might also play a role in the token generation process and pass rate. * **Error Bars:** All data points have relatively small error bars in both dimensions, suggesting that the measured means are fairly stable for each specific configuration of `n_p` and `n_fr`. * **Unused Legend Entry:** The legend includes `n_p = 25` (dark blue line segment), but no data points corresponding to this `n_p` value are plotted on the chart. * **Maximum Pass Rate:** The "Mean pass rate" never exceeds approximately 0.16-0.17, even at the highest token generation counts shown. ### Interpretation This chart likely illustrates the performance of a system or model, where `n_p` and `n_fr` are parameters influencing its output. The "Mean number of tokens generated" could represent the computational effort or output length, while "Mean pass rate" is a measure of success or quality. The overall trend suggests that increasing the number of tokens generated initially improves the pass rate, but there are diminishing returns. Beyond a certain point (around 4000-5000 tokens), generating more tokens does not significantly increase the pass rate, which plateaus at a relatively low value (around 15-16%). This could imply that the system reaches its performance limit or that generating more tokens beyond this threshold does not add value in terms of correctness or quality. The different colored and shaped points indicate that specific combinations of `n_p` and `n_fr` lead to different outcomes. For instance, `n_p=10` (bright blue circle) achieves one of the highest pass rates with a moderate number of tokens, while `n_p=2` (goldenrod triangle) at a very high token count (~8350) results in a lower pass rate than some points with fewer tokens. This suggests a complex interaction between `n_p`, `n_fr`, and the resulting performance. It's not simply "more tokens = better pass rate" across all parameter settings. The absence of data points for `n_p=25` might indicate that experiments for this parameter value were not conducted, or perhaps they yielded results outside the displayed range or were deemed irrelevant. The low overall pass rates (maxing out below 0.2) suggest that the task being evaluated is challenging, or the system's performance is generally limited under these conditions. The confidence interval around the main trend line provides a sense of the variability or uncertainty in the average performance across all observed conditions.

\n ## Line Chart: Mean Pass Rate vs. Mean Number of Tokens Generated ### Overview This chart displays the relationship between the mean number of tokens generated and the mean pass rate, for different values of *n_p* and *n_fr*. The chart uses line plots with error bars to represent the data, and includes a legend to differentiate between the different values of *n_p*. The x-axis represents the mean number of tokens generated, ranging from 0 to 10000. The y-axis represents the mean pass rate, ranging from 0 to 1.0. ### Components/Axes * **X-axis Title:** "Mean number of tokens generated" (Scale: 0 to 10000, linear) * **Y-axis Title:** "Mean pass rate" (Scale: 0 to 1.0, linear) * **Legend:** Located in the top-right corner. * *n_p* = 1 (Brown line) * *n_p* = 2 (Orange line) * *n_p* = 5 (Light Green line) * *n_p* = 10 (Blue line) * *n_p* = 25 (Purple line) * *n_fr* = 1 (Gray circle) * *n_fr* = 3 (Red triangle) * *n_fr* = 5 (Black square) * *n_fr* = 10 (Gray triangle) * **Gridlines:** Present to aid in reading values. ### Detailed Analysis The chart shows several lines representing different values of *n_p*. Each line has associated error bars, indicating the variability in the data. The lines generally show a slight upward trend, but the increase is minimal. The data points are sparse, making precise value extraction difficult. Here's an approximate extraction of data points, noting the uncertainty due to the visual nature of the chart and the error bars: * **n_p = 1 (Brown):** * At 0 tokens: ~0.05 pass rate * At 2000 tokens: ~0.10 pass rate * At 4000 tokens: ~0.12 pass rate * At 8000 tokens: ~0.14 pass rate * **n_p = 2 (Orange):** * At 0 tokens: ~0.05 pass rate * At 2000 tokens: ~0.12 pass rate * At 4000 tokens: ~0.14 pass rate * At 8000 tokens: ~0.15 pass rate * **n_p = 5 (Light Green):** * At 0 tokens: ~0.07 pass rate * At 2000 tokens: ~0.14 pass rate * At 4000 tokens: ~0.16 pass rate * At 8000 tokens: ~0.17 pass rate * **n_p = 10 (Blue):** * At 0 tokens: ~0.08 pass rate * At 2000 tokens: ~0.15 pass rate * At 6000 tokens: ~0.18 pass rate * At 8000 tokens: ~0.19 pass rate * **n_p = 25 (Purple):** * At 0 tokens: ~0.07 pass rate * At 2000 tokens: ~0.14 pass rate * At 4000 tokens: ~0.16 pass rate * At 8000 tokens: ~0.18 pass rate The gray circles, red triangles, black squares, and gray triangles represent *n_fr* values of 1, 3, 5, and 10 respectively. These points are scattered across the chart and do not form clear lines. Their values are difficult to extract precisely due to their small size and the presence of error bars. ### Key Observations * The mean pass rate generally increases with the mean number of tokens generated, but the increase is small. * Higher values of *n_p* tend to result in slightly higher mean pass rates. * The error bars indicate significant variability in the data, making it difficult to draw strong conclusions. * The *n_fr* data points appear to be independent of the *n_p* lines. ### Interpretation The chart suggests that increasing the number of tokens generated has a limited impact on the mean pass rate. The parameter *n_p* appears to have a more noticeable, though still modest, effect. The variability in the data, as indicated by the error bars, suggests that other factors may be influencing the pass rate. The scattered *n_fr* points suggest that this parameter does not have a strong correlation with either the number of tokens generated or the pass rate, or that the relationship is obscured by the variability in the data. The chart likely represents the performance of a system (e.g., a language model) as it generates text. *n_p* could represent a parameter related to prompt engineering or decoding strategy, while *n_fr* could represent a parameter related to filtering or evaluation. The data suggests that optimizing *n_p* may lead to slight improvements in performance, but that simply generating more tokens does not necessarily guarantee a higher pass rate. Further investigation would be needed to understand the underlying reasons for the observed variability and to identify other factors that may be influencing the pass rate.

## Scatter Plot with Trend Line: Mean Pass Rate vs. Mean Tokens Generated ### Overview The image is a technical scatter plot chart displaying the relationship between the "Mean number of tokens generated" (x-axis) and the "Mean pass rate" (y-axis). The chart includes multiple data series differentiated by two parameters: `n_p` (represented by color) and `n_fr` (represented by marker shape). A dark gray trend line with a light gray shaded confidence interval runs through the data points. ### Components/Axes * **X-Axis:** Labeled "Mean number of tokens generated". The scale runs from 0 to 10,000, with major tick marks at 0, 2000, 4000, 6000, 8000, and 10000. The labels are rotated approximately 45 degrees. * **Y-Axis:** Labeled "Mean pass rate". The scale runs from 0.0 to 1.0, with major tick marks at 0.0, 0.2, 0.4, 0.6, 0.8, and 1.0. * **Legend (Top-Right Quadrant):** A two-column legend box defines the data series. * **Left Column (Color for `n_p`):** * Burnt Orange line: `n_p = 1` * Golden Yellow line: `n_p = 2` * Teal line: `n_p = 5` * Sky Blue line: `n_p = 10` * Dark Navy Blue line: `n_p = 25` * **Right Column (Marker Shape for `n_fr`):** * Circle (●): `n_fr = 1` * Inverted Triangle (▼): `n_fr = 3` * Square (■): `n_fr = 5` * Triangle (▲): `n_fr = 10` * **Data Series:** Each data point is a combination of a color (from `n_p`) and a marker shape (from `n_fr`), with vertical error bars. * **Trend Line:** A solid dark gray line with a surrounding light gray shaded area (likely representing a confidence interval or standard error band). ### Detailed Analysis **Data Points (Approximate Values, reading left to right):** 1. **~500 tokens, ~0.04 pass rate:** Burnt Orange Circle (`n_p=1`, `n_fr=1`). 2. **~1200 tokens, ~0.06 pass rate:** Golden Yellow Inverted Triangle (`n_p=2`, `n_fr=3`). 3. **~2200 tokens, ~0.07 pass rate:** Burnt Orange Square (`n_p=1`, `n_fr=5`). 4. **~2800 tokens, ~0.09 pass rate:** Golden Yellow Inverted Triangle (`n_p=2`, `n_fr=3`). 5. **~3200 tokens, ~0.11 pass rate:** Teal Circle (`n_p=5`, `n_fr=1`). 6. **~4200 tokens, ~0.08 pass rate:** Burnt Orange Triangle (`n_p=1`, `n_fr=10`). 7. **~4500 tokens, ~0.11 pass rate:** Golden Yellow Square (`n_p=2`, `n_fr=5`). 8. **~6300 tokens, ~0.15 pass rate:** Sky Blue Circle (`n_p=10`, `n_fr=1`). 9. **~7000 tokens, ~0.15 pass rate:** Teal Inverted Triangle (`n_p=5`, `n_fr=3`). 10. **~8200 tokens, ~0.13 pass rate:** Golden Yellow Triangle (`n_p=2`, `n_fr=10`). **Trend Line:** The dark gray trend line starts near (0, 0.03), rises steeply until approximately 2000-3000 tokens, and then continues to rise at a much slower, almost linear rate, reaching approximately 0.15 at 10,000 tokens. The shaded confidence band is narrowest at the start and widens slightly as the token count increases. ### Key Observations 1. **Overall Trend:** There is a positive correlation between the mean number of tokens generated and the mean pass rate. The relationship appears logarithmic or asymptotic, showing strong initial gains that diminish as token count increases. 2. **Parameter Influence:** The data points for higher `n_p` values (Teal `n_p=5`, Sky Blue `n_p=10`) are positioned further to the right (higher token counts) and generally higher on the y-axis (higher pass rates) than points for lower `n_p` values (Burnt Orange `n_p=1`, Golden Yellow `n_p=2`). 3. **Marker (`n_fr`) Distribution:** For a given `n_p` color, different `n_fr` markers are spread along the x-axis. For example, the Golden Yellow (`n_p=2`) series includes an inverted triangle at ~1200 tokens, a square at ~4500 tokens, and a triangle at ~8200 tokens. 4. **Plateau Effect:** The trend line suggests the mean pass rate begins to plateau around 0.15-0.16 after approximately 6000 tokens, indicating diminishing returns for generating additional tokens beyond this point. 5. **Variability:** The error bars on individual data points indicate variability in the mean pass rate measurement for each specific parameter combination. ### Interpretation This chart likely evaluates the performance of a text generation or code synthesis system, where "pass rate" is a metric of success (e.g., passing unit tests). The "mean number of tokens generated" is a measure of output length or computational cost. The data suggests a fundamental trade-off: allowing the system to generate more tokens (a longer output) increases its probability of success, but with sharply diminishing returns. The initial increase in pass rate is significant, but after a certain point (~6000 tokens in this context), additional tokens yield minimal improvement. The parameters `n_p` and `n_fr` appear to be control variables for the generation process. The trend that higher `n_p` values are associated with both longer generations and higher pass rates implies that `n_p` might be a parameter like "number of proposals" or "parallel attempts," where allocating more resources (`n_p`) leads to better but more verbose solutions. The `n_fr` parameter, varied within each `n_p` color group, might represent a different aspect of the process, like "number of refinement steps," whose effect on token count and pass rate is less straightforward from this visualization alone. The plateau in the trend line is a critical insight for system optimization. It indicates an operational sweet spot: generating beyond ~6000-8000 tokens may waste computational resources for negligible gain in success rate. The chart provides empirical evidence for setting efficient generation limits.

## Heatmap: Mean Pass Rate vs. Mean Number of Tokens Generated ### Overview The heatmap illustrates the mean pass rate of a system as a function of the mean number of tokens generated. The x-axis represents the mean number of tokens generated, while the y-axis represents the mean pass rate. ### Components/Axes - **X-axis**: Mean number of tokens generated (ranging from 0 to 10,000) - **Y-axis**: Mean pass rate (ranging from 0.0 to 1.0) - **Legend**: - **Brown circles**: n_p = 1 - **Yellow triangles**: n_p = 2 - **Green squares**: n_p = 5 - **Blue diamonds**: n_p = 10 - **Purple stars**: n_p = 25 - **Black dots**: n_fr = 1 - **Orange triangles**: n_fr = 3 - **Blue squares**: n_fr = 5 - **Green circles**: n_fr = 10 - **Purple triangles**: n_fr = 25 ### Detailed Analysis or ### Content Details The heatmap shows that as the mean number of tokens generated increases, the mean pass rate generally increases as well. However, the rate of increase is not uniform across all categories. For example, the system with n_p = 1 and n_fr = 1 (brown circles) shows a relatively stable pass rate, while the system with n_p = 25 and n_fr = 25 (purple stars) shows a slight decrease in pass rate as the number of tokens generated increases. ### Key Observations - The system with n_p = 1 and n_fr = 1 (brown circles) has the lowest mean pass rate and the most stable performance. - The system with n_p = 25 and n_fr = 25 (purple stars) has the highest mean pass rate and the most variability in performance. - The systems with n_p = 2 and n_fr = 3 (yellow triangles and orange triangles) show a moderate increase in pass rate as the number of tokens generated increases. - The systems with n_p = 5 and n_fr = 5 (green squares and blue diamonds) show a slight increase in pass rate as the number of tokens generated increases. - The systems with n_p = 10 and n_fr = 10 (blue diamonds and green circles) show a slight increase in pass rate as the number of tokens generated increases. ### Interpretation The heatmap suggests that the system's performance is influenced by both the number of tokens generated and the number of tokens received. The system with the highest number of tokens generated (n_p = 25) and the highest number of tokens received (n_fr = 25) shows the best performance, while the system with the lowest number of tokens generated (n_p = 1) and the lowest number of tokens received (n_fr = 1) shows the worst performance. The systems with moderate numbers of tokens generated and received show a balanced performance. The slight increase in pass rate as the number of tokens generated increases suggests that the system is able to handle more tokens, but the rate of increase is not uniform across all categories.

## Line Graph: Mean Pass Rate vs. Mean Number of Tokens Generated ### Overview The image is a line graph depicting the relationship between the **mean pass rate** (y-axis) and the **mean number of tokens generated** (x-axis). Multiple data series are plotted, each corresponding to different combinations of parameters `n_p` (number of participants) and `n_fr` (number of free tokens). The graph includes shaded regions representing confidence intervals and a legend in the top-right corner. --- ### Components/Axes - **X-axis**: "Mean number of tokens generated" (ranges from 0 to 10,000 in increments of 2,000). - **Y-axis**: "Mean pass rate" (ranges from 0.0 to 1.0 in increments of 0.2). - **Legend**: Located in the top-right corner, with the following entries: - `n_p = 1` (red circle) - `n_p = 2` (orange triangle) - `n_p = 5` (green square) - `n_p = 10` (blue diamond) - `n_p = 25` (dark blue line) - `n_fr = 1` (red circle) - `n_fr = 3` (orange triangle) - `n_fr = 5` (green square) - `n_fr = 10` (blue diamond) - **Shaded Regions**: Gray bands around each line, indicating confidence intervals. --- ### Detailed Analysis #### Data Series Trends 1. **`n_p = 1` (red circle)**: - At 2,000 tokens: ~0.05 pass rate. - At 4,000 tokens: ~0.07 pass rate. - At 6,000 tokens: ~0.09 pass rate. - At 8,000 tokens: ~0.11 pass rate. - At 10,000 tokens: ~0.13 pass rate. - **Trend**: Gradual increase with diminishing returns. 2. **`n_p = 2` (orange triangle)**: - At 2,000 tokens: ~0.07 pass rate. - At 4,000 tokens: ~0.09 pass rate. - At 6,000 tokens: ~0.11 pass rate. - At 8,000 tokens: ~0.13 pass rate. - At 10,000 tokens: ~0.15 pass rate. - **Trend**: Slightly steeper than `n_p = 1`. 3. **`n_p = 5` (green square)**: - At 4,000 tokens: ~0.11 pass rate. - At 6,000 tokens: ~0.13 pass rate. - At 8,000 tokens: ~0.15 pass rate. - At 10,000 tokens: ~0.17 pass rate. - **Trend**: Faster growth than lower `n_p` values. 4. **`n_p = 10` (blue diamond)**: - At 6,000 tokens: ~0.15 pass rate. - At 8,000 tokens: ~0.17 pass rate. - At 10,000 tokens: ~0.19 pass rate. - **Trend**: Steeper than `n_p = 5`. 5. **`n_p = 25` (dark blue line)**: - At 8,000 tokens: ~0.18 pass rate. - At 10,000 tokens: ~0.20 pass rate. - **Trend**: Highest pass rate, with minimal increase at higher token counts. 6. **`n_fr = 1` (red circle)**: - Matches `n_p = 1` data points. 7. **`n_fr = 3` (orange triangle)**: - Matches `n_p = 2` data points. 8. **`n_fr = 5` (green square)**: - Matches `n_p = 5` data points. 9. **`n_fr = 10` (blue diamond)**: - Matches `n_p = 10` data points. #### Confidence Intervals - Shaded regions around each line indicate variability. For example: - `n_p = 1` has the widest confidence interval (e.g., ±0.02 at 2,000 tokens). - `n_p = 25` has the narrowest confidence interval (e.g., ±0.01 at 10,000 tokens). --- ### Key Observations 1. **Positive Correlation**: Higher `n_p` values generally correspond to higher mean pass rates. 2. **Diminishing Returns**: Pass rates plateau as the number of tokens increases, especially for lower `n_p` values. 3. **Confidence Intervals**: Larger `n_p` values (e.g., 25) show tighter confidence intervals, suggesting more reliable estimates. 4. **Parameter Relationship**: `n_fr` values are directly tied to `n_p` (e.g., `n_fr = 1` for `n_p = 1`, `n_fr = 10` for `n_p = 10`). --- ### Interpretation The data suggests that increasing the number of participants (`n_p`) improves the mean pass rate, but the effect diminishes as the number of tokens grows. The shaded confidence intervals highlight that higher `n_p` values (e.g., 25) provide more precise estimates, likely due to reduced sampling variability. The direct mapping of `n_fr` to `n_p` implies a designed relationship between free tokens and participant count, possibly to balance resource allocation. The plateau effect at higher token counts indicates a saturation point where additional tokens yield minimal performance gains.