Maybe the fates got clumsy. Maybe Poseidon had one of his angry days. However it happened, the underworld is overcrowded, and Zeus has ordered Hades to let some spirits out. Hades arranges all the souls of the dead in a line before Cerberus. When one of his three heads bites down on the soul in front of it, they’ll get returned to the land of the living. Anyone to the left must get out of line and stay in Hades forever. And everyone else shuffles forward, at which point Cerberus will feed again. Each of the dog’s heads has an equal chance of being the one to bite each time, and no two ever bite simultaneously.
也许是命运的捉弄。 也许是波塞冬生气的一天。 然而事情还是发生了, 地狱里的人太多了, 宙斯于是吩咐哈迪斯释放一些鬼神。 哈迪斯让所有亡灵在 刻耳柏洛斯面前排成一排。 当三头狗的其中一个头 咬了它面前的亡灵的时候, 这个亡灵将会回到 他们生活的土地上。 任何位于左边的亡灵 都要离开队伍,永远留在地狱。 其他亡灵就要继续前进, 三头狗就继续吃掉面前的亡灵。 三头狗的三个头 咬亡灵的概率都是均等的, 只有一个头能够伸出来咬亡灵。
Unfortunately, Hades’ minions forgot to tell you what was happening, and by the time you show up there are only 99 souls left in line. Hades looks furious and drawing attention to yourself won’t end well. But suddenly, time freezes, and Hermes steps out of the shadows. He tells you he can instantly put you into the line, and no one will realize what happened. But he’ll only grant his grace to someone clever enough to take full advantage of it. Choose the best place in line and he’ll give you the spot. Choose wrong, and he’ll leave you to rot.
不幸的是,哈迪斯的仆人 忘记告诉你这个规则, 当你知道发生了什么的时候 队伍里只有 99 个亡灵了。 哈迪斯非常生气,把注意力 吸引到你身上可不会有好结果。 但是突然,时间停止了, 赫尔墨斯从影子里走出来。 他告诉你,他可以 马上将你排入队伍里, 并且没有人会发现发生了什么。 但是他只把这个恩典 赐予足够聪明, 能够充分利用这个机会的人。 只有选择了队伍中最佳的位置, 他才会把这个位置给你。 选择错误,你就只能等死。
Which spot should you pick?
【哪一个位置是你应该选择的呢?】
Pause the video to figure it out yourself.
【暂停视频,思考一下吧】
Answer in 3
【答案揭晓 3 】
Answer in 2
【答案揭晓 2 】
Answer in 1
【答案揭晓 1 】
It’s possible to calculate the exact probability of going free in all 100 spots. But there’s a much simpler path to the solution that requires surprisingly little calculation.
在 100 个位置中 计算出精确的逃脱机率 是有可能的。 但是这里有更简单的解决方法, 几乎不需要进行任何计算。
Imagine being anywhere in line. Way up at the front, one of the three heads will pick someone at random, and you’ll move forward 1, 2, or 3 spaces. Since each is equally likely, your chance of survival from wherever you started is the average of the chances from each of the three spaces in front of you.
想象你在队伍中的任意一个位置。 在队伍的最前面,三头狗的 三个头将随机地咬掉一个亡灵, 你将会向前移动一个、 两个,或三个位置。 因为每一种情况都是等概率的, 所以你从起点出发的生存几率 就是你前面三个位置的 生存几率的平均值。
And this is where you can find a huge shortcut. Averages must be on or between the extremes of what you’re averaging— they can never be higher than the highest value or lower than the lowest. So whatever your chances of survival are where you start, one of the three places in front of you is at least as good, and probably better.
也就是说,你可以从中 找到一个很大的捷径。 平均数一定就是,或者 在你要平均的极值之间—— 平均数不会高于最高值, 也不会低于最低值。 所以无论在起始点 你的生存的几率是多少, 在你前面的三个位置, 至少有一个一样好, 也许更好。
This observation is incredibly powerful. It means that wherever you are in line, it’d be wise to trade your place for one of the three spots in front of you. Let’s ignore which for now and think of them as a trio— this trio’s maximum value is better than this trio’s, and so on. Keep going and you’ll reach the front... These three spots must contain the extreme values— the best and worst probabilities— for the entire line. In other words, they’re all we need to consider.
这一观察结果具有强大的力量。 这意味着, 无论你在队伍的哪个位置, 用你的位置去交换前面 三个位置中的一个都是明智的。 让我们先忽略要选哪个位置,然后 把这些位置以三个为一组进行思考—— 这三个的最高值比这三个高, 以此类推。 继续向前走,你将会来到最前面...... 这三个位置一定包括了 整个队伍中的极值, 即最好和最差的概率。 换句话说,这些就是 我们需要思考的全部数值。
Place 1 is bad. Head one would save you, and the other two doom you forever. That’s just a 1 in 3 chance to escape.
位置 1 并不好。 第一个头将会拯救你, 其余的两个将会让你永远消失。 这里只有三分之一的机会逃离地狱。
Place 2 is better: head two is great, head 3 is bad, and head 1 is ok in that it gives you another chance.
位置 2 不错:第二个头很好, 第三个头不好,第一个头一般般, 因为它会给你第二个机会。
But place 3 is best, because head 3 saves you while heads 1 and 2 both give you extra chances.
但是第三个位置是最好的, 因为第三个头会拯救你, 而第一个头和第二个头 只给了你额外的机会。
If you did want to consider the exact probabilities, the odds of surviving in place 3 are 16 out of 27, or close to 60%. The spots later in line tend to be very close to having a 50% chance of survival. Why 50%? Because every time Cerberus sends one soul up to be reborn, he leaves 0, 1, or 2 souls in the underworld. That averages out to one person staying for each one who gets freed.
如果你不想去思考准确的概率。 位置 3 的生存机率是 16 / 27, 或者说接近 60%。 队伍后面的位置的 生存机率无限接近于 50%。 为什么是 50%? 因为每一次三条狗 都让一位亡灵复活, 再让 0 个,1 个或者 2 个亡灵留在地狱。 平均下来,一个人被释放, 就有一个人留下来。
But you can beat those odds handily with what you now know. Hermes has places to be, and so do you. He rewards your insight by sneaking you into the third spot. And from there it’ll be just a short wait to learn your ultimate fate.
但是利用目前已知的信息, 你已经可以轻易地战胜这些机率。 赫耳墨斯有它的位置, 你也一样。 他安排你前往第三个位置, 以奖励你的洞察力。 在那里,只需要短暂的一瞬, 你就会得知自己最后的归宿。