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.
也許命運變得笨拙。 也許波賽頓經歷了憤怒的一日。 不論是如何發生的, 冥界變得十分擁擠。 於是,宙斯命令冥王 哈迪斯釋放一些靈魂。 哈迪斯讓所有死去的靈魂在 地獄犬面前排成一隊。 當牠三個頭中的一個 咬住牠面前的靈魂時, 這些靈魂就會重返人間。 在其左側的必須離開隊伍, 永遠留在冥界。 所有其他人則向前推進, 此時地獄犬將再次進食。 每次,每個頭都有同等的咬食機會, 而且遠永不會有兩個頭同時咬食。
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,哈迪斯的僕從 忘記告訴你正在發生的這件事, 等你露面時,只剩下 99 個靈魂在排隊。 哈迪斯看上去很生氣, 而且引起他注意自己不會有好下場。 突然時間停止了。 赫耳墨斯從陰影裡走了出來。 他對你說,他可以 把你立即放入隊伍中, 而且不會有人意識到發生了什麼。 但是,他只將恩典賜予足夠聰明, 可以充分利用它的人。 如果選擇了隊伍中的最佳位置, 他就會把那個位置給你。
Which spot should you pick?
如果選錯了,他則會讓你自生自滅。
Pause the video to figure it out yourself.
你會選擇哪個位置?
Answer in 3
【暫停視頻,自行解題】 【答案揭曉,3 秒】
Answer in 2
【答案揭曉,2 秒】
Answer in 1
【答案揭曉,1 秒】
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.
對這 100 個位置的獲釋概率 一一加以計算是有可能的。 但是,有一條更為簡單的解決途徑, 它幾乎不需要什麼計算。 想像站在隊伍裡的任何一個地方。
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.
在隊伍的前方,三個頭中的 一個將隨機挑選一個人, 之後你要向前移動 1,2,或者 3 個位置。 由於可能性均等, 因此無論從哪裡開始,你的倖存機會 都會是你前面三個位置的平均機會。 就在這裡你可以發現一個大捷徑。
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.
平均數必須是均值, 或者處在被平均的極值之內—— 絕不可能高於最高值 或低於最低值。 因此,無論你起始 位置的倖存機率如何, 你前面三個位置中的一個 至少與你的位置一樣好, 甚至可能更好。 這一觀察非常有力。
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.
這就意味著無論你在 隊伍中的哪個位置上, 明智之舉是將你的位置換成 前面三個位置中的一個。 讓我們暫且忽略具體是哪一個, 而將它們視為一個三件組合—— 這個組合的最大值 好過這個組合,依此類推。 繼續下去,你將到達隊伍的最前面。 這三個位置中必定包含了極值—— 即在整個隊伍中的最佳和最差機率, 換句話說,是我們需要考慮的全部。 第 1 個位置很糟。
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.
第 1 個頭會放過你, 而另兩個將使你萬劫不復, 也就是逃生機會只有三分之一。 第 2 個位置好一些。
Place 2 is better: head two is great, head 3 is bad, and head 1 is ok in that it gives you another chance.
第 2 個頭不錯, 第 3 個頭很糟, 而第 1 個頭還可以,給你另一次機會。
But place 3 is best, because head 3 saves you while heads 1 and 2 both give you extra chances.
但是第 3 個位置最好。 因為第 3 個頭放過你, 同時第 1 個和第 2 個頭 給你多出來的機會。
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.
如果你確實想知道準確概率的話, 那麼第 3 個位置的倖存機率是 27 分之 16,或接近於 60%。 隊伍後面的位置基本具有 將近 50% 的倖存機會。 為什麼是 50%? 因為每當地獄犬送一個靈魂去重生, 就會在冥界中留下 0、1 或 2 個靈魂。 平均每釋放一個人就會 有另一個人被留下。
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.
但是利用你現在所知道的, 你可以輕鬆克服這些難關。 赫耳墨斯還有事要做,你也一樣。 他將獎勵你的洞見, 把你偷偷插入到第三個位置上。 從那裡起,無需久等, 你便會知道最終的命運。