I was asked this question in an interview, and did not initially answer correctly though I still think my interpretation may have been the correct one. The question was:
我在接受采访时被问到这个问题，并且最初没有正确回答，尽管我仍然认为我的解释可能是正确的。问题是：
There are two delivery trucks, A and B. A makes deliveries between 8am and 10am, and B makes deliveries between 9am and 11am. The deliveries are uniformly distributed for both. What is the probability that any given delivery from B will take place before any delivery from A?
有两辆运货卡车A和B.A在早上8点到早上10点之间交货，B在早上9点到11点之间交货。交付均匀分配给两者。在从A交付之前，B的任何给定交付的概率是多少？
What is your answer, and why? Thanks in advance.
你的答案是什么？为什么？提前致谢。
2 个解决方案
#1
5
It's 1/8. See the figure below, which shows A's delivery time on the x-axis and B's on the y-axis. Since deliveries are uniformly distributed, all points in the square are equally likely to occur. B delivers before A only in the shaded region, which is 1/8 of the total figure.
这是1/8。请参见下图，其中显示了A轴在x轴上的传送时间，而B轴在y轴上的传送时间。由于交付均匀分布，因此广场中的所有点都可能同样发生。 B仅在阴影区域之前传递A，这是总数的1/8。
Another way to think of it is that there's a 50% chance A delivers before B even starts, and 50% chance that B delivers after A is done, meaning there's a 75% chance of one or both of those happening. In the 25% chance they both deliver in the overlapping hour, it's a 50-50 chance of which delivers first.
另一种思考方式是，在B开始之前A有50％的可能性，A在A完成后有50％的几率，这意味着其中一个或两个都有75％的可能性发生。他们都有25％的几率在重叠小时内交付，这是50％的机会首先提供。
#2
2
Since the delivery rates are not specified, lets assume A delivers $a$ packages per hour and B delivers $b$ packages per hour. So there are $2a \cdot 2b$ pairs of delivery times. The window in which A and B overlap in deliver times has only $a \cdot b$ pairs, in half of which A comes before B. So the proportion of pairs in which A comes before B is $$ \frac{a\cdot b}{2}\frac{1}{2a \cdot 2b} = \frac{1}{8}. $$
由于没有指定交付率，我们假设A每小时提供$ a $包，而B每小时提供$ b $包。所以有$ 2a \ cdot 2b $对的交货时间。 A和B在交付时间内重叠的窗口只有$ a \ cdot b $对，其中一半A在B之前。因此A在B之前出现的对的比例是$$ \ frac {a \ cdot b} {2} \ frac {1} {2a \ cdot 2b} = \ frac {1} {8}。 $$
#1
5
It's 1/8. See the figure below, which shows A's delivery time on the x-axis and B's on the y-axis. Since deliveries are uniformly distributed, all points in the square are equally likely to occur. B delivers before A only in the shaded region, which is 1/8 of the total figure.
这是1/8。请参见下图，其中显示了A轴在x轴上的传送时间，而B轴在y轴上的传送时间。由于交付均匀分布，因此广场中的所有点都可能同样发生。 B仅在阴影区域之前传递A，这是总数的1/8。
Another way to think of it is that there's a 50% chance A delivers before B even starts, and 50% chance that B delivers after A is done, meaning there's a 75% chance of one or both of those happening. In the 25% chance they both deliver in the overlapping hour, it's a 50-50 chance of which delivers first.
另一种思考方式是，在B开始之前A有50％的可能性，A在A完成后有50％的几率，这意味着其中一个或两个都有75％的可能性发生。他们都有25％的几率在重叠小时内交付，这是50％的机会首先提供。
#2
2
Since the delivery rates are not specified, lets assume A delivers $a$ packages per hour and B delivers $b$ packages per hour. So there are $2a \cdot 2b$ pairs of delivery times. The window in which A and B overlap in deliver times has only $a \cdot b$ pairs, in half of which A comes before B. So the proportion of pairs in which A comes before B is $$ \frac{a\cdot b}{2}\frac{1}{2a \cdot 2b} = \frac{1}{8}. $$
由于没有指定交付率，我们假设A每小时提供$ a $包，而B每小时提供$ b $包。所以有$ 2a \ cdot 2b $对的交货时间。 A和B在交付时间内重叠的窗口只有$ a \ cdot b $对，其中一半A在B之前。因此A在B之前出现的对的比例是$$ \ frac {a \ cdot b} {2} \ frac {1} {2a \ cdot 2b} = \ frac {1} {8}。 $$