Investigation 1 - Tossing Drawing Pins | 探究1：抛图钉实验

📚 Investigation 1 - Tossing Drawing Pins | 探究1：抛图钉实验

Probability is often introduced through equally likely outcomes, such as tossing a fair coin or rolling a die. This investigation challenges that assumption by examining the behaviour of drawing pins, which are not symmetrical and therefore do not produce outcomes with equal probability. We begin with a hands-on experiment, move from observed frequencies to theoretical models, and then critically assess whether a binomial distribution fits the data using a formal goodness-of-fit test. The investigation mirrors the kind of exploratory work students undertake in their IB Mathematics Internal Assessment, blending statistical reasoning, experimental design, and reflective evaluation.

概率通常通过等可能结果来引入，例如抛掷一枚公平的硬币或掷骰子。本次探究通过检验图钉的行为来挑战这一假设，因为图钉形状不对称，其结果概率并不相等。我们从动手实验开始，从观测频数过渡到理论模型，然后使用拟合优度检验严格评估二项分布是否与数据吻合。这一探究过程与学生在 IB 数学内部评估中所进行的探索性工作相似，融合了统计推理、实验设计和反思性评价。

1. The Drawing Pin as a Random Object | 图钉作为随机对象

Unlike a coin, a drawing pin has a flat head and a sharp point, so when tossed onto a hard surface, it can land in one of two distinct orientations: point up (pin rests on its head with the point in the air) or point down (pin rests with its point touching the surface and head tilted). Because of the asymmetrical shape, the two outcomes are not equally likely. This makes the drawing pin an excellent vehicle for investigating experimental probability and for testing a binomial model where the probability of success is unknown.

与硬币不同，图钉有一个扁平的头和一个尖锐的针尖，因此当它抛掷到坚硬的表面上时，会以两种截然不同的朝向落下：针尖向上（图钉以头部为支撑，针尖在空中）或针尖向下（针尖接触表面，头部倾斜）。由于形状不对称，这两种结果并不等可能。这使得图钉成为探究实验概率以及检验成功概率未知的二项模型的绝佳工具。

2. Setting Up the Experiment | 实验设置

For the investigation, a single type of drawing pin is selected to keep the physical properties consistent. The pin is tossed from a fixed height onto a flat, hard floor, and only tosses that result in a clear 'point up' or 'point down' are recorded; any ambiguous landings (e.g. against a wall) are discarded and retossed. A fixed number of tosses, say 10, constitutes one trial, and this trial is repeated many times—typically 50 or 100 trials—to build a frequency distribution for the number of 'point up' outcomes per trial.

在本次探究中，选用单一类型的图钉以保持物理性质一致。将图钉从固定高度抛向平坦的硬地面，仅记录针尖明显“朝上”或“朝下”的结果；任何模棱两可的落地（如靠墙）都予以舍弃并重新抛掷。每 10 次抛掷构成一次试验，将这一试验重复多次——通常为 50 或 100 次——以构建每次试验中“针尖朝上”次数的频数分布。

3. Collecting and Organising Data | 收集与整理数据

Suppose 100 trials of 10 tosses each are performed. For each trial, the number of 'point up' results is counted. These values range from 0 to 10. The observed frequencies are then tallied in a table. For instance, the data might show that 0 point-up outcomes occurred in 1 trial, 1 point-up in 3 trials, 2 in 8 trials, and so on. This table becomes the foundation for calculating the experimental probability of each outcome and the overall proportion of success for a single toss.

假设进行了 100 次试验，每次试验抛掷 10 次。对每次试验，记录“针尖向上”的次数。这些取值从 0 到 10 不等。然后将观测频数列入表格。例如，数据可能显示 0 次针尖向上出现了 1 次试验，1 次向上出现 3 次试验，2 次向上出现 8 次试验，以此类推。这张表格便成为计算每种结果的实验概率以及单次抛掷成功比例的基础。

4. Estimating the Probability of Success | 估计成功概率

From the raw data, the total number of 'point up' d. If results across all trials is computed. If there are 100 trials of 10 tosses, there are 1000 individual tosses. Let the total number of point-up outcomes be, for example, 420. The estimated probability of a single toss landing point up, denoted p̂ te serves as the parameter for the binomial model we will later test. It is crucial to note that this p̂ is a point estimate derived from the experiment and is subject to sampling variability.

根据原始数据，计算所有试验中“针尖向上”结果的总数。若进行了 100 次试验，每次 10 次抛掷，则共有 1000 次单独抛掷。假设针尖向上的总次数为 420。那么单次抛掷针尖向上的估计概率 p̂ 为 420/1000 = 0.42。这一估计值将作为随后检验的二项模型的参数。必须注意，p̂ 是由实验得出的点估计值，并且存在抽样变异。

5. Building a Binomial Model | 构建二项模型

If each toss of the drawing pin is independent and the probability of landing point up is constant, then the number of successes X in 10 tosses follows a binomial distribution: X ~ B(10, p̂). The theoretical probability of exactly k successes is given by

P(X = k) = ¹⁰Cₖ · (p̂)ᵏ · (1 − p̂)¹⁰⁻ᵏ

其中 ¹⁰Cₖ 是二项式系数，等于 10! / (k! (10−k)!)。Using p̂ = 0.42, we can compute the probability for k = 0, 1, …, 10.

如果图钉的每次抛掷相互独立，且针尖向上的概率恒定，那么 10 次抛掷中成功的次数 X 服从二项分布：X ~ B(10, p̂)。恰好获得 k 次成功的理论概率由下式给出

P(X = k) = ¹⁰Cₖ · (p̂)ᵏ · (1 − p̂)¹⁰⁻ᵏ

其中 ¹⁰Cₖ 是二项式系数。利用 p̂ = 0.42，我们可以计算 k = 0, 1, …, 10 的概率。

6. Comparing Observed and Expected Frequencies | 比较观测频数与期望频数

d is 8. Visual comparison through a bar chart can reveal any obvious patterns, but a statistical test is needed to decide whether the differences are due to chance alone or indicate that the binomial model is inadequate.

在得知理论概率后，我们将每个概率乘以试验总次数（100），得到期望频数。然后构建一张观测频数与期望频数并排的表格。例如，k = 2 的期望频数可能在 10.4 左右，而观测值为 8。通过条形图进行视觉比较可以揭示明显的模式，但需要通过统计检验来判断差异是否仅仅由偶然因素造成，还是表明二项模型不适用。

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7. Chi-Squared Goodness-of-Fit Test | 卡方拟合优度检验

The chi-squared (χ²) test is an appropriate tool here. The test statistic is:

χ² = Σ (O − E)² / E

e observed frequency and E the expected frequency for each category (k = 0 to 10). Since p̂ = 9. The null hypothesis states that the data come from a binomial distribution with p = p̂. If the calculated χ² exceeds the critical value at a significance level of 5%, we reject the binomial model.

卡方检验（χ²）是这里恰当的工具。检验统计量为：

χ² = Σ (O − E)² / E

其中 O 代表每个类别（k = 0 到 10）的观测频数，E 为期望频数。由于 p̂ 是从数据中估计的，自由度为（类别数 − 1 − 估计参数个数）= 11 − 1 − 1 = 9。原假设为数据来自参数 p = p̂ 的二项分布。若计算得到的 χ² 在 5% 显著性水平下超过临界值，则拒绝二项模型。

8. Assumptions of the Binomial Model | 二项模型的假设

For the binomial distribution to be valid, four conditions must hold: a fixed number of tosses per trial, only two possible outcomes per toss, a constant probability of success, and independence between tosses. While the first two are clearly satisfied, the constancy of p and independence are more questionable. For instance, tossing a large number of pins might cause wear or deformation, slightly altering the probability. Additionally, if the same pin is tossed repeatedly, the outcome of one toss could physically affect the next if the pin is not perfectly reset, though this is usually negligible. These assumptions need to be discussed in the evaluation.

要使二项分布成立，必须满足四个条件：每次试验的抛掷次数固定，每次抛掷只有两种可能结果，成功概率恒定，以及抛掷之间相互独立。前两个条件显然满足，但 p 的恒定性和独立性则更值得商榷。例如，大量抛掷图钉可能导致磨损或变形，轻微改变概率。此外，如果反复抛掷同一枚图钉，一次抛掷的结果可能会在物理上影响下一次（如果图钉没有完全复位），不过这通常可以忽略。这些假设需要在评估中加以讨论。

9. Interpreting the Test Results | 解读检验结果

If the chi-squared test produces a p-value greater than 0.05, we do not have sufficient evidence to reject the binomial model, which suggests that the observed variation is consistent with random sampling from B(10, p̂). In this case, we can conclude that the experimental data support the use of a binomial distribution for the tossing drawing pins experiment. If, however, the p-value is very small, we must consider possible violations of the independence or constant-p assumptions, or perhaps the existence of other landing categories that were ignored.

如果卡方检验产生的 p 值大于 0.05，则没有充分证据拒绝二项模型，这表明观测到的变异与从 B(10, p̂) 中随机抽样一致。在这种情况下，我们可以得出结论：实验数据支持在抛图钉实验中使用二项分布。然而，如果 p 值非常小，我们就必须考虑可能违反了独立性或 p 恒定的假设，或者可能存在其他被忽略的落地类别。

10. Possible Alternative Models | 可能的替代模型

If the binomial fit is poor, an alternative approach is to treat the number of tosses until the first 'point up' as a geometric random variable. We could also consider that the probability p might drift over time due to handling, leading to a beta-binomial model that allows for overdispersion. Furthermore, a multinomial model might be considered if the landing is classified into more than two states (e.g. point up, point down, pin on its side). These extensions offer rich avenues for further investigation and align well with the IA criterion for mathematical depth.

如果二项拟合不佳，另一种思路是将首次出现“针尖向上”所需的抛掷次数视为几何随机变量。我们还可以考虑，由于操作处理，概率 p 可能随时间飘移，从而采用允许过度离散的贝塔-二项模型。此外，如果落地状态分为两类以上（例如针尖向上、针尖向下、图钉侧躺），则可以考虑多项分布模型。这些拓展为进一步的探究提供了丰富的途径，并且与 IA 对数学深度的要求非常契合。

11. Evaluation of Experimental Design | 实验设计的评价

No experiment is perfect. In this investigation, potential sources of error include inconsistent tossing technique, insufficient sample size leading to low expected frequencies in some categories (which inflates the χ² statistic), and the subjective judgment of 'point up' in borderline landings. To improve reliability, one could use a mechanical tossing device, increase the number of trials, or pool categories with expected counts below 5. Reflecting on these limitations is an essential part of the mathematical exploration and demonstrates critical thinking.

没有实验是完美的。在本次探究中，可能的误差来源包括：抛掷手法不一致，样本量不足导致某些类别的期望频数偏低（这会膨胀 χ² 统计量），以及在对介乎两者之间的落地结果进行“针尖向上”判断时的主观性。为提高可靠性，可以使用机械抛掷装置，增加试验次数，或将期望频数低于 5 的类别进行合并。对这些局限性的反思是数学探索的重要组成部分，并体现出批判性思维。

12. Extending the Investigation and Conclusion | 拓展探究与结论

This investigation establishes a framework for testing a probability model against real data. Students can extend the work by varying the surface (carpet vs. wood), the type of pin, or the number of tosses per trial to see if the estimated p changes. They might also explore whether the geometric distribution fits the waiting time for the first 'point up'. Tossing a simple drawing pin opens a gateway to statistical hypothesis testing, experimental design, and the philosophy of modeling — all central themes in the IB Mathematics course. The process reinforces that probability is not merely a theoretical exercise but a deeply empirical discipline.

本次探究构建了一个用真实数据检验概率模型的框架。学生可以通过改变表面（地毯与木地板）、图钉类型或每次试验的抛掷次数来拓展研究，看估计的 p 是否变化。还可以探索几何分布是否能很好地拟合首次“针尖向上”的等待时间。抛掷一枚简单的图钉，开启了通往统计假设检验、实验设计以及建模哲学的大门——而这些都是 IB 数学课程的核心主题。这一过程强化了一个认识：概率不仅仅是理论推演，更是一门高度经验性的学科。

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