幸运之门

The Lucky Door

专题: Probability / 概率
难度: L3
来源: OpenQuant

第 1 小问

题目详情

你正在参加一个游戏节目，并被要求选择幸运门。一扇门后面是 10,000 美元。剩下的四人什么也没有。你预计能赢得多少奖金？

You're on a gameshow and are tasked to choose the lucky door. Behind 1 door is $10,000. The remaining four have nothing. How much are you expected to win?

解析

你选择正确门的概率是 $\frac15$ 。当你选择门时，你的奖金是10,000。因此你对这款游戏的预期收益为 $\frac15 \cdot 10,000 = 2,000$

Original Explanation

The probaility you select the correct door is $\frac15$ . Your prize when you select the door is 10,000. Thus your expected earnings of this game is $\frac15 \cdot 10,000 = 2,000$

第 2 小问

题目详情

你有机会购买一副 X 射线护目镜，它可以让你在电池耗尽之前准确地看清一扇门。你愿意花多少钱购买这些护目镜？

You have the opportunity to purchase a pair of x-ray goggles that will allow you to see through exactly 1 door before the battery runs out. How much are you willing to pay for these goggles?

解析

如果你使用X射线护目镜，则有一个 $\frac15$ ，你透过门看到钱，而一个 $\frac45$ ，你有机会知道你选择的门没有钱。在后一种情况下，你知道以该价格选择 $4$ 剩余门中的一扇。使用此信息，你可以计算使用护目镜的预期奖金：

\frac15 \cdot 10,000 + \frac45 \cdot \frac14 \cdot10,000 = 4,000

你的预期奖金增加了 $2,000$ ，因此你应该愿意支付高达 2000 美元购买 X 射线护目镜。

Original Explanation

If you use the x-ray goggles, there is a $\frac15$ you look through a door and see the money and a $\frac45$ chance you know the door you selected has no money. In the latter case, you know to select one of the $4$ remaining doors for the price. Using this information you can compute expected winnings with the goggles:

\frac15 \cdot 10,000 + \frac45 \cdot \frac14 \cdot10,000 = 4,000

Your expected winnings has increased by $2,000$ so you should be willing to pay up to $2000 for the x-ray goggles.

第 3 小问

题目详情

使用 X 光护目镜后，你会发现门后什么也没有。你选择 $4$ 剩余门中的一扇。在这里，游戏节目主持人会奖励你惊人的数学技能，并打开一扇背后什么也没有的门。他让你可以选择坚持当前的猜测或切换到任何其他门。假设你使用最优策略，你获胜的概率是多少？

After using the x-ray goggles you see that there is nothing behind the door you looked through. You select one of the $4$ remaining doors. Here is where the game show host rewards you for your amazing math skills and opens a door which also has nothing behind it. He gives you the option to stick with your current guess or switch to any other door. What is the probability of you winning assuming you use optimal strategy?

解析

使用护目镜，你可以消除 $1$ 或 $5$ 门。这意味着你选择的新门有 $\frac14$ 的机会获得奖品。这也意味着钱有 $\frac34$ 的机会进入剩余的一扇门。主人给你展示了剩下的一扇没有钱的门。钱在剩下的一扇门后面的概率仍然是 $\frac34$ ，但分布在 $2$ 门之间，使任一门有 $\frac38$ 成为幸运门的机会。

Original Explanation

Using the goggles you were able to eliminate $1$ of the $5$ doors. This means the new door you pick has a $\frac14$ chance of having the prize. This also means there is a $\frac34$ chance that the money is in one of the remaining doors. The host shows you a remaining door which does not have the money. The probability the money is behind one of the remaining doors is still $\frac34$ , but is spread between $2$ doors giving either door a $\frac38$ chance of being the lucky door.