Can you solve the basketball riddle? - Dan Katz

翻译人员: Su Liu 校对人员: Lexi Ding

You’ve spent months creating a basketball-playing robot,

你花了几个月的时间 创造了一款篮球机器人，

the Dunk-O-Matic,

给它取名为“灌篮高手”。

and you’re excited to demonstrate it at the prestigious Sportecha Conference.

你很高兴能在著名的 体育大会上展示它。

Until you read an advertisement:

直到你读到一则广告：

“See the Dunk-O-Matic face human players and automatically adjust its skill

“快来看灌篮高手对战人类玩家， 它能在场上自动做出调整，

to create a fair game for every opponent!”

为每个对手创造公平的比赛！”

That's not what you were told to create.

这跟说好的可不一样。

You designed a robot that shoots baskets,

你设计的机器人

sometimes successfully and sometimes not, taking turns with a human opponent.

在和人类对手轮流投篮时， 有时能投进有时投不进。

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

没有人说过要教它调整性能。

Maybe the CEO skimmed an article about AI and overpromised,

也许是总裁浏览了一篇 有关 AI 的文章，夸下了海口，

setting you up for public embarrassment.

导致你要当众丢脸。

Luckily, you installed a feature where given any probability q,

幸运的是，你植入了一项功能， 在给定任意概率 q 的情况下，

you can adjust the robot to have that probability of success on each attempt.

你能调整机器人，使其每次 投篮命中的概率都达到 q。

You swiftly gather information, and jackpot:

你迅速收集信息， 太走运了：

your team has a dossier on all potential demo participants,

你的团队有一份档案，记录了 所有将要与机器人对战的对手信息，

including the probability each has of making baskets.

包括每个人投篮命中的概率。

In each match, the human shoots first, then the robot, then the human again,

在每场比赛中，人类玩家先投， 机器人后投，然后人类玩家再投，

and so on until someone makes the first successful basket and wins.

以此类推，直到一方 成功命中，赢得胜利。

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

你可以远程调整灌篮高手 对战不同对手时的命中概率。

What should that probability be for each opponent,

机器人的命中概率是多少时，

so that the human has a 50% chance of winning each match?

人类玩家获胜的机率为 50%呢？

Pause here to figure it out yourself. Answer in 3

暂停一下，自己想一想。 3 秒之后揭晓答案

Answer in 2

2 秒

Answer in 1

1 秒

You might guess that q should be equal to p.

你可能会猜 q 应该等于 p。

But that ignores the advantage of going first.

但这忽略了先手优势。

Suppose p and q are both 100%.

假设 p 和 q 都是 100%。

Even though the competitors are equally skilled,

尽管竞争对手同样熟练，

the first player always wins.

但总是先投的获胜。

So a deeper analysis is required.

因此，我们需要更深入的分析。

One approach involves adding up every chance the human has to win,

一种方法是利用几何级数

using geometric series.

将人类获胜的每一次机会加起来。

A geometric series is an infinite sum of numbers,

几何级数是数字的无限相加，

where each number is the previous number multiplied by a common ratio.

其中每个数字都是 前一个数字乘以公比。

Two facts about geometric series are useful here.

有两个关于几何级数的规律很有用。

First, if the common ratio r of a geometric series

首先，如果几何级数的公比 r

has absolute value less than 1,

绝对值小于 1，

the series has a finite total.

那么这个数列的总数是有限的。

And second, if the first number in the series is a,

第二，如果数列中的 第一个数字是 a，

that total is: a divided by 1-minus-r.

则总数为：a /（1 - r）。

How does this help us calibrate our robot?

这对我们校准机器人有什么用？

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

人类投篮命中的概率为 p。

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

由于人类先投，那么他们 第一次投篮就获胜的概率为 p。

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

他们第二次投篮获胜的概率是多少？

That attempt only happens if both players miss.

只有两个玩家都没投进， 才会进行第二次投篮。

The probability of a miss is 1 minus the probability of a success,

失误概率为 1 减去成功概率，

so the miss probabilities are 1-minus-p and 1-minus-q.

因此失败概率为 1 - p 和 1 - q。

The chance of both happening is the product of those values.

二者相乘就是双方都没投进的概率。

So the probability of two failures and then a human success

因此，双方第一轮投篮失败 然后人类第二轮成功的概率为

is p times (1-minus-p) times (1-minus-q).

p * (1 - p) * (1 - q)

Winning on the third try requires another round of misses,

在第三轮中获胜则需要再失误一轮，

so that chance is p multiplied by the double-miss probability twice.

因此概率为 p 乘以 双方两次失误的概率。

If we add all the possible probabilities of a human win,

如果我们将人类获胜的 所有可能概率相加，

the total is the sum of a geometric series.

则总数是几何级数的总和。

Since the first number in the series is p,

由于数列中的第一个数字是 p，

and the ratio is this product that’s less than 1,

而公比是小于 1 的乘积，

the sum will be (p divided by 1) minus the ratio.

因此总和为 p 除以（1 - 公比）。

We want this sum to be 1/2.

我们希望这个总和是 1/2。

Using some algebra to solve for q,

使用代数方法求解 q，

we find that q should equal p divided by 1-minus-p.

我们发现 q = p / (1 - p)。

If p is greater than 50%, q would need to be bigger than 1,

如果 p 大于 50%， 则 q 需要大于 1，

which can’t happen.

而这不可能。

In that case, a fair game is impossible,

在这种情况下，比赛不可能公平，

because the human has a better-than-50% chance of winning immediately.

因为人类第一轮投篮 就获胜的机率超过 50％。

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

机器人获胜的总概率 也是几何级数的总和。

How does this series compare to the human’s?

这个数列与人类的相比如何？

To win, the robot needs some number of double misses,

机器人要想获胜， 比赛中双方要先失误几次，

then a human failure followed by a robot success.

接着人类再次失误， 而随后机器人投篮命中。

If q equals p over 1-minus-p,

如果 q = p / (1 - p)，

(1-minus-p) times q is p.

那么（1 - p）* q = p。

For our choice of q, not only do these series have the same sum,

不管 q 的值是多少， 数列不仅具有相同的和，

but they’re the same series!

整个数列也完全相同！

We could bypass geometric series by starting with this reasoning.

从这个推理出发， 我们可以绕过几何级数。

The robot’s chances of winning in the first round is (1-minus-p) times q,

机器人在第一轮中获胜的概率是 （1 - p）* q，

and so if we want that chance to match the human’s first-round chance,

因此，如果我们想让这个概率 与人类首轮获胜的机会相同，

we want it to equal p, making q: p over 1-minus-p.

即我们希望它等于 p， 那么 q = p /（ 1 - p）。

More rounds may occur, but before each round,

可能会有更多轮比赛， 但在每轮比赛之前，

the competitors are tied, so everything effectively restarts.

参赛者都会打成平手， 因此一切实际上都会重新开始。

If they have the same odds of winning in the first round,

如果他们在第一轮比赛中的胜率相同，

they also will in the second round, and so on.

那么他们在第二轮比赛中的胜率 也会相同，以此类推。

The demonstration goes perfectly,

展示进行得很顺利，

but while you didn't want to embarrass yourself,

尽管你不想让自己感到尴尬，

you also didn’t want to deceive the public.

但是你也不想欺骗公众。

Taking the stage, you explain your company’s false promises

上台后，你解释了公司的虚假承诺

and your hastily ad-libbed solution.

和你匆忙制定的解决方案。

Thankfully, the ensuing bad press is directed at your employers,

值得庆幸的是，随之而来的负面新闻 是针对你的雇主的，

and it turns out the presentation volunteers

原来演讲志愿者们

own a more employee-friendly robotics company.

拥有一家对员工更友好的机器人公司。

After some tedious intellectual property litigation,

在经历了一些乏味的知识产权诉讼后，

you find yourself at a healthier workplace

你发现自己的工作环境更加健康，

with a regular spot on a pickup basketball team.

还在临时篮球队中赢得一席之地。