Create a fair game You've spent months creating a basketball-playing robot, the Dunk-O-Matic, and you're excited to demonstrate it at the prestigious Sportecca conference.

創造一個公平的遊戲 你花了幾個月時間創造了一個會打籃球的機器人 Dunk-O-Matic，你很想在著名的 Sportecca 會議上展示它。

Until you read an advertisement, see the Dunk-O-Matic face human players and automatically adjust its skill to create a fair game for every opponent.

直到您看到廣告，看到 Dunk-O-Matic 面對人類玩家，自動調整技能，為每個對手創造公平的比賽。

That's not what you were told to create.

這不是要你創作的東西。

You designed a robot that shoots baskets, sometimes successfully and sometimes not, taking turns with a human opponent.

你設計了一個機器人，與人類對手輪流投籃，有時成功，有時失敗。

No one said anything about teaching it to adjust its performance.

沒有人說過要教它調整性能。

Maybe the CEO skimmed an article about AI and overpromised, setting you up for public embarrassment.

也許首席執行官只是略讀了一篇關於人工智能的文章，就做出了過高的承諾，讓你當眾出醜。

Luckily, you installed a feature where, given any probability Q, you can adjust the robot to have that probability of success on each attempt.

幸運的是，您安裝了一項功能，只要給定任意概率 Q，您就可以調整機器人，使其每次嘗試的成功概率都達到該概率。

You swiftly gather information and jackpot.

您迅速收集資訊並中獎。

Your team has a dossier on all potential demo participants, including the probability each has of making baskets.

你的團隊有一份所有潛在演示參與者的檔案，包括每個人上籃的概率。

In each match, the human shoots first, then the robot, then the human again, and so on until someone makes the first successful basket and wins.

在每場比賽中，人類先投籃，然後機器人投籃，最後人類再投籃，依此類推，直到有人投中第一個成功的籃筐並獲勝。

You can remotely adjust the Dunk-O-Matic's probability between opponents.

您可以遠程調整 Dunk-O-Matic 在不同對手之間的概率。

What should that probability be for each opponent so that the human has a 50% chance of winning each match?

為了讓人類在每場比賽中都有 50% 的勝算，每個對手的勝算概率應該是多少？

Pause here if you want to figure it out for yourself.

如果您想自己弄明白，請在此暫停。

Answer in 3 Answer in 2 Answer in 1 You might guess that Q should be equal to P, but that ignores the advantage of going first.

答案為 3 答案為 2 答案為 1 你可能會猜 Q 應該等於 P，但這忽略了先走的優勢。

Suppose P and Q are both 100%.

假設 P 和 Q 都是 100%。

Even though the competitors are equally skilled, the first player always wins, so a deeper analysis is required.

儘管競爭對手的技術水平相當，但第一位選手總是獲勝，是以需要進行更深入的分析。

One approach involves adding up every chance the human has to win using geometric series.

其中一種方法是利用幾何級數將人類獲勝的每一次機會加起來。

A geometric series is an infinite sum of numbers where each number is the previous number multiplied by a common ratio.

幾何級數是一個無限的數字總和，其中每個數字都是前一個數字乘以一個共同的比率。

Two facts about geometric series are useful here.

關於幾何級數的兩個事實在這裡很有用。

First, if the common ratio R of a geometric series has absolute value less than 1, the series has a finite total.

首先，如果一個幾何數列的公比 R 的絕對值小於 1，則該數列的總數是有限的。

And second, if the first number in the series is A, that total is A divided by 1 minus R.

其次，如果數列中的第一個數字是 A，那麼總數就是 A 除以 1 再減去 R。

How does this help us calibrate our robot?

這對我們校準機器人有什麼幫助？

Remember that the human has probability P of making a basket.

請記住，人類上籃的概率為 P。

Since they go first, they have probability P of winning on the first try.

由於他們先走，他們第一次嘗試獲勝的概率為 P。

What's the probability that they win on the second try?

他們在第二次嘗試中獲勝的概率是多少？

That attempt only happens if both players miss.

只有在雙方都失誤的情況下，才會發生這種嘗試。

The probability of a miss is 1 minus the probability of a success, so the miss probabilities are 1 minus P and 1 minus Q.

失誤概率是 1 減去成功概率，是以失誤概率是 1 減去 P 和 1 減去 Q。

The chance of both happening is the product of those values, so the probability of two failures and then a human success is P times 1 minus P times 1 minus Q.

這兩種情況發生的概率是這些值的乘積，是以，兩次失敗和一次成功的概率是 P 乘以 1 減去 P 乘以 1 再減去 Q。

Winning on the third try requires another round of misses, so that chance is P multiplied by the double-miss probability twice.

要在第三次嘗試中獲勝，需要再進行一輪失誤，是以這個概率是 P 乘以兩次雙失誤概率。

If we add all the possible probabilities of a human win, the total is the sum of a geometric series.

如果我們把人類獲勝的所有可能概率加起來，總數就是一個幾何數列的總和。

Since the first number in the series is P, and the ratio is this product that's less than 1, the sum will be P divided by 1 minus the ratio.

由於數列中的第一個數是 P，而比值是小於 1 的乘積，所以總和將是 P 除以 1 再減去比值。

We want this sum to be one-half.

我們希望這個總數是二分之一。

Using some algebra to solve for Q, we find that Q should equal P divided by 1 minus P.

用代數方法求 Q，我們發現 Q 應該等於 P 除以 1 再減去 P。

If P is greater than 50%, Q would need to be bigger than 1, which can't happen.

如果 P 大於 50%，Q 就必須大於 1，而這是不可能的。

In that case, a fair game is impossible, because the human has a better-than-50% chance of winning immediately.

在這種情況下，公平遊戲是不可能的，因為人類有超過 50% 的機會立即獲勝。

The robot's total probability is also the total of a geometric series.

機器人的總概率也是幾何級數的總和。

How does this series compare to the human's?

這個系列與人類的系列相比如何？

To win, the robot needs some number of double misses, then a human failure, followed by a robot success.

要想獲勝，機器人需要一定數量的兩次失誤，然後人類失敗，接著機器人成功。

If Q equals P over 1 minus P, 1 minus P times Q is P.

如果 Q 等於 P 超過 1 減 P，那麼 1 減 P 乘以 Q 就是 P。

For our choice of Q, not only do these series have the same sum, but they're the same series.

對於我們選擇的 Q，這些數列不僅具有相同的和，而且是相同的數列。

We could bypass geometric series by starting with this reasoning.

我們可以從這個推理出發，繞過幾何級數。

The robot's chances of winning in the first round is 1 minus P times Q.

機器人在第一輪獲勝的概率是 1 減 P 乘以 Q。

And so if we want that chance to match the human's first-round chance, we want it to equal P, making Q P over 1 minus P.

是以，如果我們想讓這一概率與人類的首輪概率相匹配，我們就想讓它等於 P，從而使 Q P 大於 1 減 P。

More rounds may occur, but before each round, the competitors are tied, so everything effectively restarts.

可能會有更多輪比賽，但在每輪比賽之前，參賽者都會打成平手，是以一切實際上都會重新開始。

If they have the same odds of winning in the first round, they also will in the second round, and so on.

如果他們在第一輪比賽中的勝率相同，那麼他們在第二輪比賽中的勝率也會相同，以此類推。

The demonstration goes perfectly, but while you didn't want to embarrass yourself, you also didn't want to deceive the public.

演示進行得非常順利，但你不想讓自己難堪，同時也不想欺騙公眾。

Taking the stage, you explain your company's false promises and your hastily ad-libbed solution.

上臺後，你解釋了公司的虛假承諾和你匆忙編造的解決方案。

Thankfully, the ensuing bad press is directed at your employers, and it turns out the presentation volunteers own a more employee-friendly robotics company.

值得慶幸的是，隨之而來的負面新聞是針對你的僱主的，原來演講志願者擁有一家對員工更友好的機器人公司。

After some tedious intellectual property litigation, you find yourself at a healthier workplace, with a regular spot on a pickup basketball team.

在經歷了一些乏味的知識產權訴訟後，你發現自己的工作環境更健康了，還經常參加籃球隊。

Intuitively Visualized Courses, and that's just one among thousands of STEM lessons.

直觀可視化課程》，而這只是數以千計的 STEM 課程中的一門。

Visit Brilliant.org slash TED-Ed for 30 days free and 20% off an annual subscription.

訪問 Brilliant.org slash TED-Ed，享受 30 天免費服務和 20% 的年度訂閱折扣。