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抛一枚公平的硬币时,出现正面或反面的概率通常被理解为 50-50。这个概念基于一个关键假设:硬币的两面出现的可能性相同,因此每种结果(正面或反面)发生的可能性都是 50%。
Have you ever lost a coin toss? In theory, the odds could have gone either way. The stakes may or may not have been low for you, but millions of dollars are regularly wagered on the outcome of the Super Bowl coin toss. But what if a coin lands on its side, rather than either heads or tails? Then who emerges victorious?
你曾经丢过硬币吗?从理论上讲,两种情况都有可能发生。对你来说,赌注可能很低,也可能不低,但人们经常为超级碗掷硬币的结果下注数百万美元。但是,如果硬币落地时是侧面朝上,而不是正面或反面呢?那么谁会取得胜利呢?
In this mini-lecture, L. Mahadevan, de Valpine professor of applied mathematics, organismic and evolutionary biology, and physics, explores the probability of a coin toss in which the coin lands on its side, using it as a metaphor to discuss the role of chance and mathematics in understanding the world around us.
在这个迷你讲座中,de Valpine 应用数学、有机体和进化生物学以及物理学教授 L. Mahadevan 探讨了抛硬币时硬币侧着地的概率,并用它作为隐喻来讨论硬币的作用机会和数学在理解我们周围的世界中的作用。
The probability of getting heads or tails when tossing a fair coin is often understood as 50-50. This notion is based on a critical assumption: the two sides of the coin are equally likely, and therefore each outcome—heads or tails—has a 50 percent chance of occurring.
抛一枚公平的硬币时,出现正面或反面的概率通常被理解为 50-50。这个概念基于一个关键假设:硬币的两面出现的可能性相同,因此每种结果(正面或反面)发生的可能性都是 50%。
Building on the simple coin toss, imagine gluing multiple coins together into a stack. As the stack becomes taller, the chance of landing on its side increases significantly, changing the probability dynamics completely.
在简单的抛硬币的基础上,想象一下将多个硬币粘在一起形成一叠。随着堆栈变得更高,侧面着陆的机会显着增加,完全改变概率动态。
As the stack grows taller, approaching a mile long, the likelihood of the coin landing on its side approaches nearly 100 percent. Thus, the probability of it landing on heads or tails reduces dramatically, until gravity defines it as zero.
随着硬币堆越来越高,接近一英里长,硬币侧面落地的可能性接近 100%。因此,它正面或反面着陆的概率大大降低,直到重力将其定义为零。
As Mahadevan explains, it’s possible to create a coin where the probability of heads, tails, and landing on its side are equal. About eight coins glued together to form a stack will produce an equal probability for all three scenarios, with approximately a one-third chance of each outcome.
正如 Mahadevan 所解释的那样,制造出正面、反面和侧面落地的概率相等的硬币是可能的。大约八枚硬币粘在一起形成一堆,对于所有三种情况都会产生相同的概率,每种结果的概率约为三分之一。
On one hand, the trajectory of a coin in flight is governed by Newton’s laws of motion, which can be precisely predicted. On the other, uncertainty—whether the coin lands heads, tails, or on its side—is an inherent part of the process, determined by initial conditions and the interplay of forces such as gravity and air resistance. By understanding and manipulating parameters such as shape, size, and mass, we can influence outcomes—effectively “tilting the odds” in our favor.
一方面,硬币飞行的轨迹受牛顿运动定律控制,可以精确预测。另一方面,不确定性——硬币是正面、反面还是侧面——是该过程的固有部分,由初始条件以及重力和空气阻力等力的相互作用决定。通过理解和操纵形状、大小和质量等参数,我们可以影响结果——有效地“扭转局面”,对我们有利。
In a larger sense, this observation invites us to consider how deterministic laws and randomness co-exist in nature. Although the laws of physics always hold, we can sometimes wield an understanding of these rules to achieve a desired outcome—an outcome that might seem impossible or improbable.
从更大的意义上来说,这一观察结果让我们思考确定性规律和随机性在自然界中是如何共存的。尽管物理定律始终成立,但我们有时可以运用对这些规则的理解来实现预期的结果——一个看似不可能或不太可能的结果。
In his course General Education 1190: “Wonder Why: Science as a Culture of Curiosity,” last offered in the spring of 2024, Mahadevan encouraged his students to think critically and to question the assumptions underlying what they observe. He assigned them an experiment to investigate the relationship between the length of a cylindrical object and the probability of it landing on its side.
Mahadevan 在 2024 年春季最后开设的通识教育 1190 课程:“想知道为什么:科学作为一种好奇心文化”中,鼓励学生进行批判性思考,并对他们所观察到的现象背后的假设提出质疑。他给他们布置了一个实验,研究圆柱形物体的长度与其侧面着陆的概率之间的关系。
By varying the length of rods and conducting numerous trials, the students gathered thousands of data points, which were then compared to theoretical predictions. The experimental results closely aligned with the theory, emphasizing the power of mathematics to describe the world accurately.
通过改变杆的长度并进行多次试验,学生们收集了数千个数据点,然后将其与理论预测进行比较。实验结果与理论紧密结合,强调了数学准确描述世界的力量。
In the sonnet that Mahadevan shared with his students, he used poetry to capture the essence of these ideas. The humble coin, traveling in “parabolic flight” as dictated by Newton’s laws, becomes a symbol of the delicate balance between certainty and uncertainty.
在马哈德万与学生分享的十四行诗中,他用诗歌来捕捉这些思想的精髓。这枚不起眼的硬币按照牛顿定律进行“抛物线飞行”,成为确定性与不确定性之间微妙平衡的象征。
“See the humble coin in parabolic flighton a path prescribed by Newton's laws.While outcomes flicker in the clearest lightUncertainty reigns and gives one pause.
“看看一枚不起眼的硬币沿着牛顿定律规定的抛物线飞行。虽然结果在最清晰的光线中闪烁,但不确定性占主导地位并让人停顿。
A cylinder long may land upon its side“I wonder why,” pondered students' minds.The ratio of width to the diameter does abide,Which von Neumann cleverly into thirds assigned.
一个长圆柱体可能会侧着地着陆“我想知道为什么,”学生们思考着。宽度与直径的比率确实存在,冯·诺依曼巧妙地将其划分为三分之一。
But chance is not maximally blind,And angular momentum unlocks the key.In precessions where biases unwind,The coin's true fate might not what it seems be.
但机会并不是完全盲目的,而角动量解锁了钥匙。在偏见消除的进动中,硬币的真正命运可能并不像看上去的那样。
From the Gen Ed view, every motion we discern,Shows magic where physics and chance turn.”
从 Gen Ed 的角度来看,我们察觉到的每一个动作,都展现了物理和机会转变的魔力。”
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