📚 Revisiting Drawing Pins: A Statistical Investigation | 重新审视图钉实验：一项统计调查
A classic probability experiment involves tossing a drawing pin and observing whether it lands with its point facing upwards or downwards. This seemingly simple random process opens the door to fundamental statistical ideas: from modelling with the binomial distribution and estimating an unknown probability, to testing hypotheses and evaluating experimental design. In this investigation, we revisit the drawing pin experiment with a modern statistical lens, complementing hand-collected data with simulation, and critically examining the assumptions that underpin our conclusions.
一个经典的概率实验是抛掷一枚图钉，观察它落地时是钉尖朝上还是朝下。这个看似简单的随机过程开启了一扇通往统计学基本思想的大门：从用二项分布建模、估计未知概率，到假设检验和评估实验设计。在本次调查中，我们用现代的统计学视角重新审视图钉实验，将手动收集的数据与模拟相结合，并批判性地审视支撑我们结论的假设。
1. Introduction to the Drawing Pin Experiment | 图钉实验介绍
A standard drawing pin, also known as a thumbtack, is not a symmetric object. Unlike a fair coin, it does not have two equally likely outcomes when flipped. The mass is concentrated in the flat head, and the pin's aerodynamic properties cause it to land with the point up more or less frequently depending on its shape, the surface, and the method of dropping. This makes the drawing pin an ideal vehicle for exploring real-world probability estimation and statistical inference, as the true probability of landing 'pin up' is unknown and must be inferred from data.
一枚普通的图钉并不是对称物体。与一枚均匀的硬币不同，它在抛掷时并不会出现两个等可能的结果。质量集中在扁平的钉帽上，加之其空气动力学特性，使得它落地时钉尖朝上的频率会因其形状、落地表面和抛掷方式而高低不同。这使得图钉成为探索真实世界中概率估计与统计推断的理想载体，因为“钉尖朝上”的真实概率是未知的，必须通过数据来推断。
The experiment is typically conducted by dropping a drawing pin from a fixed height onto a hard, flat surface and recording the number of 'pin up' and 'pin down' outcomes after a series of trials. This setup generates binary data suitable for a Bernoulli or binomial model, and the investigation can be scaled from a simple point estimate of the success probability to full inferential procedures, including confidence intervals and goodness-of-fit tests.
该实验通常是将一枚图钉从固定高度下落到一个坚硬、平整的表面上，并在若干次试验后记录“钉尖朝上”和“钉尖朝下”的结果。这样的设置会生成适合伯努利或二项模型的二元数据，而这次调查可以从成功概率的简单点估计一直深入到包括置信区间和拟合优度检验在内的完整推断方法。
2. Setting Up the Investigation: Aims and Hypotheses | 调查设定：目标与假设
investigation is to estimate the probability that a particular brand of drawing pin lands 'pin up' st whether the observed data deviate significantly from a hypothesised probability of 0.5, which would represent a 'fair' thesis H₀: p = 0.5 against the alternative H₁: p ≠ 0.5, where p is the true probability of 'pin up'.
本次调查的主要目的是估计某种特定品牌的图钉在受控条件下落地时“钉尖朝上”的概率，并评估该概率是否能够在各次试验中视为恒定。次要目的是检验观测数据是否显著偏离假设的 0.5 的概率，该假设代表一枚“均匀”图钉。正式地，我们提出原假设 H₀: p = 0.5 和备择假设 H₁: p ≠ 0.5，其中 p 为“钉尖朝上”的真实概率。
Beyond the single-proportion test, the investigation will examine factors such as drop height and surface texture, extending the analysis to a comparison of two different drawing pin brands. By combining physical experimentation with calculator-based and spreadsheet simulations, we will also illustrate the impact of sample size on the precision of our estimates, bringing the law of large numbers to life.
除了单比例检验，调查还将考察下落高度和表面质地等因素，将分析拓展到两种不同品牌图钉的比较。通过将物理实验与基于计算器和电子表格的模拟相结合，我们还将展示样本量对估计精度的影响，从而将大数定律生动地展现出来。
3. Experimental Design and Data Collection | 实验设计与数据收集
To ensure consistency, the following protocol was adopted. A single brand of brass-plated drawing pin was selected, with mass 0.85 g and head diameter 8 mm. The pin was dropped from a height of 30 cm onto a laminated wooden surface, using a mechanical release mechanism to minimise human bias. The outcome was recorded as 'U' (pin up) or 'D' (pin down). A total of 200 trials were performed, with the pin carefully inspected after every 50 drops to check for deformation. The environment was kept free from drafts and vibrations.
为确保一致性，采用了以下方案。选择某一品牌的镀铜图钉，质量为 0.85 g，钉帽直径为 8 mm。图钉从 30 cm 高度自由下落到层压木板表面，使用机械释放装置以最大限度减少人为偏差。结果记录为 'U'（钉尖朝上）或 'D'（钉尖朝下）。总共进行 200 次试验，每 50 次抛掷后仔细检查图钉是否变形。环境保持无气流和振动。
The raw data yielded 138 'pin up' outcomes and 62 'pin down' outcomes. This raw proportion of 0.69 immediately suggests a strong bias away from 0.5. A summary table of the observed frequencies is presented below.
原始数据得到 138 次“钉尖朝上”和 62 次“钉尖朝下”的结果。这一原始比例 0.69 立刻表明存在显著偏离 0.5 的偏差。下面展示观测频数的汇总表。
| Outcome | 结果 | Observed Frequency | 观测频数 |
|---|---|
| Pin up (U) | 钉尖朝上 | 138 |
| Pin down (D) | 钉尖朝下 | 62 |
| Total | 总计 | 200 |
4. Modelling the Experiment: Bernoulli Trials | 实验建模：伯努利试验
If we assume that each drop is independent and that the probability of 'pin up' the number of times the pin lands point up in n drops. Under these conditions, X follows a binomial distribution with parameters n and p, written X ~ The probability mass function is given by P(X = k) = ⁿCₖ pᵏ (1-p)ⁿ⁻ᵏ for k = 0, 1, ..., n.
如果我们假设每次抛掷相互独立，且每次试验“钉尖朝上”的概率保持恒定，那么这个结果序列就可以被建模为一组伯努利试验。令随机变量 X 表示在 n 次抛掷中图钉钉尖朝上的次数。在这些条件下，X 服从参数为 n 和 p 的二项分布，记作 X ~ B(n, p)。其概率质量函数为 P(X = k) = ⁿCₖ pᵏ (1-p)ⁿ⁻ᵏ，其中 k = 0, 1, ..., n。
This model is the foundation for all subsequent inference. The assumptions of independence and constant probability are crucial. Dependence could arise if the pin lands in a manner that affects its next drop, such as bending, or if the experimenter's technique changes. We mitigated this by inspecting the pin regularly and using a mechanical dropper.
该模型是所有后续推断的基础。独立性和恒定概率这两条假设至关重要。如果图钉以某种会影响下一次抛掷的方式落地，例如发生弯曲，或者实验者的操作手法发生变化，就有可能产生相关关系。我们通过定期检查图钉并使用机械投掷器来减轻了这些影响。
5. Point Estimation of the Probability of ‘Pin Up’ | “钉尖朝上”概率的点估计
The most intuitive estimator for the unknown probability p is the sample proportion p̂ = x / n, where x is the observed number of successes. With our data, p̂ 0.69. This is an unbiased estimator of p, and its standard error can be estimated as SE(p̂) = √[p̂(1 - p̂) / n]. Substituting our values gives SE(p̂) = √[0.69 × 0.31 / 200] ≈ √0.0010695 ggests that the true probability of landing pin up is approximately 0.69, with a standard error of about 0.033.
对于未知概率 p 最直观的估计量是样本比例 p̂ = x / n，其中 x 是观测到的成功次数。根据我们的数据，p̂ = 138 / 200 = 0.69。这是 p 的一个无偏估计量，其标准误可以估计为 SE(p̂) = √[p̂(1 - p̂) / n]。代入我们的值可得 SE(p̂) = √[0.69 × 0.31 / 200] ≈ √0.0010695 ≈ 0.0327。因此，点估计表明图钉钉尖朝上的真实概率约为 0.69，标准误约为 0.033。
In the context of the binomial distribution, the expected number of 'pin up' unt exactly by construction. The point estimate is a useful summary, but it does not reflect the uncertainty associated with the finite sample size. This motivates the construction of an interval estimate.
在二项分布的背景下，如果估计的概率是正确的，则 200 次试验中“钉尖朝上”的期望次数为 200 × 0.69 = 138，由于我们是基于观测值计算的，这与观测计数完全吻合。点估计是一个有用的概括，但它并未反映有限样本量所伴随的不确定性。这促使我们去构建区间估计。
6. Constructing Confidence Intervals | 构建置信区间
A 95% the normal approximation to the binomial distribution, which is reasonable here because np̂ = 138 > 10 and n(1-p̂) = 62 > 10, the interval is given by p̂ ± z* × √[p̂(1-p̂)/n], where z* = 1.96 for 95% yields 0.69 ± 1.96 × 0.0327, giving an interval of approximately (0.626, 0.754). We are 95% confident that the true probability of 'pin up' lies within this interval.
一个关于真实比例 p 的 95% 置信区间提供了一组合理的取值范围。这里使用二项分布的正态近似是合理的，因为 np̂ = 138 > 10 且 n(1-p̂) = 62 > 10。区间由下式给出：p̂ ± z* × √[p̂(1-p̂)/n]，其中对于 95% 置信度，z* = 1.96。由此得到 0.69 ± 1.96 × 0.0327，所得区间约为 (0.626, 0.754)。我们有 95% 的信心认为“钉尖朝上”的真实概率落在这个区间内。
To check the robustness of this approximation, an alternative Wilson score interval can be employed. For our data, the Wilson interval with a 95% confidence level is (0.621, 0.751), which is very close to the Wald interval. The consistency between intervals reinforces our confidence in the inference. Notably, the confidence interval does not contain 0.5, which provides initial evidence against the 'fairness' hypothesis.
为了检验该近似的稳健性，可以采用备选的威尔逊得分区间。针对我们的数据，95% 置信水平的威尔逊区间为 (0.621, 0.751)，非常接近沃尔德区间。区间间的一致性增强了我们对推断的信心。值得注意的是，该置信区间不包含 0.5，这为反对“均匀性”假设提供了初步证据。
7. Testing the Fairness Hypothesis: Chi-Squared Goodness of Fit | “公平性”假设检验：卡方拟合优度
chi-squared goodness-of-fit test. Under the null hypothesis, the expected frequencies for 'pin up' and 'pin down' .841. Since 28.88 > 3.841, we reject the null hypothesis and conclude that the drawing pin is not fair.
我们可以使用卡方拟合优度检验对 H₀: p = 0.5 进行正式检验。在原假设下，200 次试验中“钉尖朝上”和“钉尖朝下”的期望频数均为 100。检验统计量为 χ² = Σ (Oᵢ - Eᵢ)² / Eᵢ。根据我们的数据：χ² = (138 - 100)²/100 + (62 - 100)²/100 = 1444/100 + 1444/100 = 28.88。在自由度为 1（2 个类减 1）的情况下，α = 0.05 时的临界值为 3.841。因为 28.88 > 3.841，我们拒绝原假设，并得出结论：图钉并非均匀。
a complementary check, an exact binomial test yields a two-sided p-value of approximately 1.3 × 10⁻⁵, confirming the same conclusion.
与该卡方统计量相关的 p 值极小（p < 0.001），这表明纯凭偶然观察到如此极端的偏离等频数的情况是极不可能的。这一强有力的统计证据与图钉的物理不对称性是一致的。作为补充验证，精确二项检验得出的双侧 p 值约为 1.3 × 10⁻⁵，证实了相同的结论。
8. Investigating the Effect of Surface and Drop Height | 表面和下落高度的影响探究
f 100 trials were conducted: one on a carpeted surface and another from a drop height of 60 cm (using the original wooden surface). On the carpet, 72 'pin up' outcomes were recorded (p̂ = 0.72). From the increased height, 65 'pin up' outcomes were observed (p̂ = 0.65).
一个重要的假设是概率 p 保持恒定，不受轻微环境变化的影响。为了检验这一点，我们额外进行了两组各 100 次的试验：一组在地毯表面进行，另一组从 60 cm 高度下落（使用原来的木质表面）。在地毯上，记录到 72 次“钉尖朝上”（p̂ = 0.72）。在增加高度的情况下，观察到 65 次“钉尖朝上”（p̂ = 0.65）。
A two-sample proportion z-test comparing the original 30 cm wood data (p̂₁ = 0.69) with the carpet data yields a z-statistic of z = (0.72 - 0.69) / √[p̅(1-p̅)(1/n₁ + 1/n₂)], where p̅ 0.58, suggesting no significant difference. However, comparing 30 cm and 60 cm on wood (p̂ = 0.65) gives z ≈ 0.71, p ≈ 0.48, also not significant at the 5% derate changes, though limitations in sample size for the sub-experiments should be noted.
使用双样本比例 z 检验将原始 30 cm 木板数据（p̂₁ = 0.69）与地毯数据进行比较，得到 z 统计量为 z = (0.72 - 0.69) / √[p̅(1-p̅)(1/n₁ + 1/n₂)]，其中 p̅ = (138+72)/(200+100) = 210/300 = 0.70。计算出的 z ≈ 0.55，对应的 p 值 ≈ 0.58，表明无显著差异。然而，将 30 cm 与 60 cm 木板上的数据（p̂ = 0.65）进行比较，得 z ≈ 0.71，p ≈ 0.48，在 5% 水平上同样不显著。这些结果表明，在这些适度的变化范围内，我们最初的估计是相当稳健的，但应该注意子实验样本量的局限性。
9. Sample Size and the Law of Large Numbers | 样本量与大数据定律
The precision of p̂ ive proportion of successes was plotted against the trial number. The plot showed substantial fluctuations for small trial numbers, but the cumulative proportion gradually stabilised around 0.7 as n grew.
随着样本量的增加，p̂ 的精度会提高，这是大数定律的一个结果。为了说明这一点，我们进行了一次模拟，生成了真实概率 p = 0.7 的 2000 次伯努利试验，并绘制了累积成功比例相对于试验次数的图像。图像显示，当试验次数较少时，比例波动很大，但随着 n 的增长，累积比例逐渐稳定在 0.7 附近。
The width of a confidence interval also narrows. For n = 50 with p̂ = 0.7, the approximate 95% margin of error is 1.96 × √0.7×0.3/50 drops to 0.040, yielding (0.660, 0.740). This demonstrates why larger samples yield more reliable inferences, an important consideration when designing a statistical investigation.
置信区间的宽度也会收窄。当 n = 50 且 p̂ = 0.7 时，近似 95% 误差幅度为 1.96 × √0.7×0.3/50 ≈ 0.127，所得区间约为 (0.573, 0.827)。当 n = 500 时，误差幅度降至 0.040，得到 (0.660, 0.740)。这展示了为什么更大的样本会产生更可靠的推断，这是在设计统计调查时必须考虑的重要事项。
10. Comparing Two Drawing Pin Brands | 比较两种品牌的图钉
d brand of drawing pin with a larger head and shorter point was tested using 200 drops under identical conditions (30 cm, wood). This brand produced 113 'pin up' outcomes, giving p̂ .565. A chi-squared test of homogeneity can compare the two brands. The observed table is:
为了拓展调查，我们在相同条件下（30 cm，木板）对第二种钉帽更大、钉脚更短的图钉进行了 200 次抛掷。该品牌产生了 113 次“钉尖朝上”的结果，得出 p̂₂ = 0.565。可以使用卡方同质性检验来比较这两种品牌。观测表格如下：
| Brand | 品牌 | Pin Up | 钉尖朝上 | Pin Down | 钉尖朝下 | Total | 总计 |
|---|---|---|---|
| Brand A (original) | 品牌 A | 138 | 62 | 200 |
| Brand B | 品牌 B | 113 | 87 | 200 |
| Total | 总计 | 251 | 149 | 400 |
The expected frequency for Brand A 'pin up' 6.69 > 3.841, so we reject the hypothesis that the two brands have the same probability of 'pin up'. The p-value is approximately 0.0097, indicating a significant difference.
在同质性假设下，品牌 A “钉尖朝上”的期望频数为 200 × (251/400) = 125.5。计算 χ² = Σ (O-E)²/E 得： -74.5)²/74.5 + (113-125.5)²/125.5 + (87-74.5)²/74.5 ≈ 1.245 + 2.100 + 1.245 + 2.100 = 6.69。在自由度为 1 的情况下，临界值为 3.841；χ² = 6.69 > 3.841，因此我们拒绝两个品牌具有相同“钉尖朝上”概率的假设。p 值约为 0.0097，表明存在显著差异。
11. Limitations and Extensions | 局限与扩展
While the investigation yields clear results, several limitations must be acknowledged. The assumption of independent trials may be violated if the pin experiences minor wear over 200 drops, altering its aerodynamics. Although we inspected the pin, micro-deformations could accumulate. Furthermore, the normal approximation to the binomial, though justified here, could be less accurate with smaller sample sizes or probabilities near 0 or 1.
尽管调查得出了明确的结果，但必须承认几点局限性。如果图钉在 200 次抛掷中经历轻微磨损从而改变了其空气动力学特性，那么独立试验的假设可能会被违背。尽管我们检查了图钉，微小的形变仍可能累积。此外，二项分布的正态近似虽然在本文中合理，但在样本量较小或概率接近 0 或 1 时可能不够准确。
The investigation could be extended by studying the effect of release orientation (e.g., dropping the pin point-up vs. point-down). Another rich avenue is to model the number of 'pin up' outcomes using a Bayesian approach, which would allow the incorporation of prior knowledge or expert opinion. Simulation of the physical process using physics engines could also provide insight into why p is around 0.7 for this particular design. From a statistical perspective, a logistic regression model could be built to predict the probability of 'pin up' based on head diameter, pin length, and drop height.
调查可以通过研究释放方向的影响（例如，钉尖朝上与朝下抛下）来加以拓展。另一个丰富的途径是使用贝叶斯方法对“钉尖朝上”的数量进行建模，这样便可以融入先验知识或专家意见。利用物理引擎对物理过程进行模拟，也可以为了解为何该特定设计的 p 约为 0.7 提供见解。从统计学的角度来看，可以建立一个逻辑回归模型，基于钉帽直径、钉脚长度和下落高度来预测“钉尖朝上”的概率。
12. Conclusion and Reflection | 结论与反思
investigation revisited the classic drawing pin experiment, estimating the probability of 'pin up' to be 0.69 with a 95% ), and further analysis revealed that the estimate remained stable across moderate changes in surface and height. A significant difference was found between two brands, highlighting how physical design influences the underlying probability.
本次调查重新审视了经典的图钉实验，估计出“钉尖朝上”的概率为 0.69，其 95% 置信区间为 (0.626, 0.754)。卡方检验有力地拒绝了均匀性假设（p = 0.5），进一步分析表明，该估计在表面和高度的适度变化中保持稳定。两种品牌之间发现了显著差异，突显了物理设计如何影响潜在概率。
On reflection, the drawing pin serves as a powerful pedagogical tool that bridges probability, statistics, and experimental science. The process of moving from a physical toss to a mathematical model, and then critically examining that model's assumptions, mirrors the authentic practice of data science. The most rewarding insight is that even the simplest act of dropping a pin, when viewed through a statistical lens, reveals a rich tapestry of inferential thinking.
反思来看，图钉是一个强大的教学工具，它在概率、统计和实验科学之间架起了桥梁。从物理抛掷到数学模型，再到批判性地审视该模型的假设，这一过程反映了真正的数据科学实践。最令人受益的洞见是，即使是最简单的丢一枚图钉的行为，当透过统计学的透镜观察时，也能展现出一幅精彩的推断思维画卷。
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