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拋一枚公平的硬幣時,出現正面或反面的機率通常被理解為 50-50。這個概念是基於一個關鍵假設:硬幣的兩面出現的可能性相同,因此每種結果(正面或反面)發生的可能性都是 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?
你曾經丟過硬幣嗎?從理論上講,兩種情況都有可能發生。對你來說,賭注可能很低,也可能不低,但人們經常為超級盃擲硬幣的結果下注數百萬美元。但是,如果硬幣落地時是側面朝上,而不是正面或反面呢?那麼誰會取得勝利呢?
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.
在這個迷你講座中,de Valpine 應用數學、有機體和進化生物學以及物理學教授 L. Mahadevan 探討了拋硬幣時硬幣側著地的概率,並用它作為隱喻來討論硬幣的作用機會和數學在理解我們周圍的世界中的作用。
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.
拋一枚公平的硬幣時,出現正面或反面的機率通常被理解為 50-50。這個概念是基於一個關鍵假設:硬幣的兩面出現的可能性相同,因此每種結果(正面或反面)發生的可能性都是 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.
在簡單的拋硬幣的基礎上,想像一下將多個硬幣黏在一起形成一疊。隨著堆疊變得更高,側面著陸的機會顯著增加,完全改變機率動態。
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.
隨著硬幣堆越來越高,接近一英里長,硬幣側面落地的可能性接近 100%。因此,它正面或反面著陸的機率大大降低,直到重力將其定義為零。
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 所解釋的那樣,製造出正面、反面和側面落地的機率相等的硬幣是可能的。大約八枚硬幣黏在一起形成一堆,對於所有三種情況都會產生相同的機率,每種結果的機率約為三分之一。
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,硬幣飛行的軌跡受牛頓運動定律控制,可以精確預測。另一方面,不確定性——硬幣是正面、反面還是側面——是過程的固有部分,由初始條件以及重力和空氣阻力等力的相互作用決定。透過理解和操縱形狀、大小和品質等參數,我們可以影響結果——有效地“扭轉局面”,對我們有利。
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.
從更大的意義上來說,這個觀察讓我們思考確定性規律和隨機性在自然界中是如何共存的。儘管物理定律始終成立,但我們有時可以運用對這些規則的理解來實現預期的結果——一個看似不可能或不太可能的結果。
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 年春季最後開設的通識教育 1190 課程:「想知道為什麼:科學作為好奇心文化」中,鼓勵學生進行批判性思考,並對他們所觀察到的現象背後的假設提出質疑。他給他們佈置了一個實驗,研究圓柱形物體的長度與其側面著陸的機率之間的關係。
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.
透過改變桿的長度並進行多次試驗,學生們收集了數千個數據點,然後將其與理論預測進行比較。實驗結果與理論緊密結合,強調了數學準確地描述世界的力量。
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.
在馬哈德萬與學生分享的十四行詩中,他用詩捕捉這些思想的精髓。這枚不起眼的硬幣按照牛頓定律進行“拋物線飛行”,成為確定性與不確定性之間微妙平衡的象徵。
“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幣沿著牛頓定律規定的拋物線飛行。雖然結果在最清晰的光線中閃爍,但不確定性占主導地位並讓人停頓。
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.
一個長圓柱體可能會側著地著陸「我想知道為什麼,」學生們思考著。
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.
但機會並不是完全盲目的,而角動量解鎖了鑰匙。
From the Gen Ed view, every motion we discern,Shows magic where physics and chance turn.”
從 Gen Ed 的角度來看,我們察覺到的每一個動作,都展現了物理和機會轉變的魔力。
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