科学与数学jakevdp.github.io编辑精选
等待时间悖论：为什么我的公交总是迟到？
The Waiting Time Paradox, Or, Why Is My Bus Always Late?
2018 — 2024，它一次次回到首页
黑点是上首页（≥ 50 分）的那次投稿，空心点是没冲上去的投稿，点越大分数越高。
导读
Jake VanderPlas 解释“等待时间悖论”：随机到站时，你更可能落入较长的发车间隔，所以平均等待往往不是直觉中的一半间隔；对泊松到达甚至等于平均间隔。文章用公交场景串起检查悖论、泊松过程与时刻表调度，说明“我的车总迟到”为何是一种概率错觉。
为什么经久不衰：它用一个日常困扰讲清检查悖论和无记忆性，反直觉又实用，多年后仍能帮人理解随机等待与调度。
HN 评论精选
日常双向公交与排队体验说明等待和运气并不像直觉那样对称，值得用简单计分游戏检验。
好文章。它让我想起在伦敦住的那一年，每天从西汉普斯特德的 West End Lane 坐公交去帝国理工。路两边各有一个站：一个去程，一个回程（公交从市中心到终点站后大致沿原路返回）。我们不看时刻表——高峰时太不准，而且车本来就频繁。但我们原本以为去程和回程谁先到概率各半。一段时间后我和女儿确信并非如此，于是发明了一个游戏（叫“生命游戏”）：如果我们的车先到，加1分；在我们的车来之前每过一辆去程车，减1分（常常不止一辆）。我们知道结果会略偏负，但长期应接近0。当然并没有。后来我们把游戏扩展到很多统计场景，比如超市结账时前面有N人，轮到你时数身后M人，M>N加生命分，否则减，累加M-N看自己多幸运。不过我从没做真正分析，所以很喜欢这篇文章。
herodotus查看英文原文
Nice article. It reminds me of my year living in London, and taking the bus everyday to Imperial College from West End Lane in West Hampstead. There was a stop on both sides of the road - one for the outbound bus, and one for the inbound (the bus went from central London to a terminus and then returned mostly on the same route). Now we did not use schedules - way too inaccurate at rush hour, and the busses there were pretty frequent anyway. But we did expect an even chance of the inbound bus arriving before an outbound one did. My daughter and I became convinced after a while that this was not happening, so we invented a game (which we called "The Game of Life".) When our bus (inbound) arrived first, we added 1 to our score. We subtracted 1 for every outbound bus that passed before ours arrived (there were often more than 1). We realized that the result would be slightly skewed to the negative, but we expected the outcome to be close to 0 over time. Of course it was not. Anyway we extended the game to many statistical situations. For example, you go to the checkout line at the supermarket, and there are N people in front of you. When you get to the front of the line, you count the people behind you - call that M. If M is bigger than N, you scored life points. If it is smaller, you lost some. So you add M-N to your running score, and you get an idea of how lucky you are in life. However, I never followed up with any real analysis, so I enjoyed this article.
泊松过程无记忆，已知上一块出块时间并不能缩短下一块的期望等待。
这让我想起比特币社区的一个赌局。如果比特币平均每10分钟出块，而你得知5分钟前刚有人找到块，你平均还要等多久？答案是10分钟，不是直觉的5分钟。这是无记忆过程，无论最近出了多少块，平均还要等10分钟。换句话说，等交易确认时去看最近一个块是多久前出的，以此估计下一块多快来，是错的；即使上一块是9分钟前找到，平均等待仍是10分钟。
ikken查看英文原文
This reminds me the bet in the bitcoin community [1]. If on average bitcoin blocks are produced every 10 minutes, and you learn that 5 minutes ago someone found a block, what is the average time you will wait for the next block? It turns out it's 10 minutes, not 5 minutes as you would intuitively think. (it's a memoryless process, so average expected time till block is always the same - 10 minutes - no matter how many blocks were recently found). In other words, when you're waiting for bitcoin transaction to be confirmed and go to check how long ago the most recent block was produced, in order to estimate how soon the next one will come - you're doing it wrong. Even if previous block was found 9 minutes ago, you're average waiting time for the next block is still 10 minutes. [1]. https://www.reddit.com/r/btc/comments/7rs8ko/dr_craig_s_wright_has_refused_to_pay_up_on_a_bet/
公交不可靠会让人放弃时刻表、采取随遇而安的策略，环境可预测性会影响规划习惯。
略相关：我住的小镇公交又少又稀，我从不信印刷时刻表。提前10分钟到也不保险，可能白等20分钟，因为车早到11分钟。有一半时候我决定走到下一镇（车多），结果所有本镇公交（双向）都从我身边开过，只能对着云喊。这让我对时间完全无所谓：在两个站之间走，来哪辆坐哪辆，随缘；靠经验估计到大地方要多久。我有个朋友完全不是这种思维，她很认真、完全信任时刻表（不过她主要坐火车，偏差小得多）。这让我想到关于智力/财富比例的儿童研究：环境随机，你就随机思考；环境可预测，你就会计划。
agumonkey查看英文原文
Slightly related, my ghost town had few buses and sparses. I could never rely on printed hours. If I got there 10 min earlier to be sure, I'd still never be sure I'd wait 20 min for nothing because it was 11 min early. Of course half the time if I decide to walk to the next town where buses are many, I'd see all my town buses (both ways) pass me <yell-at-cloud.png> I think it made me completely careless about time, I would just go between stops and take the first one, go with the flow. By experience I'd know the range it would take for me to reach big places around the area. I had a friend who was completely foreign to this mode of thinking, she was very dilligent and fully trusting (although she mostly used trains so a lot less divergence). It reminds me of kid studies about intelligence / wealth ratios. When you're environment is random, you think random. When it's predictable you planify.
质疑泊松过程无记忆性在公交和灯泡例子中的现实合理性，并追问真正适合建模的现象。
文章里说：泊松过程是无记忆过程，假设到达概率与距上次到达的时间完全无关；现实中运营良好的公交系统会刻意用时刻表避免这种行为——公交不会在一天中随机发车，而是按服务乘客的最佳时刻表发车。我从没真正理解任何泊松过程的例子：它们总是公交到站或灯泡烧坏，我不明白无记忆性对这些为什么合理。即使公交系统运营很差，为什么可以假设期望到达时间不随你已经等了多久而改变？什么实际现象才是被泊松过程很好建模的？
twtw查看英文原文
> a Poisson process is a memoryless process that assumes the probability of an arrival is entirely independent of the time since the previous arrival. In reality, a well-run bus system will have schedules deliberately structured to avoid this kind of behavior: buses don't begin their routes at random times throughout the day, but rather begin their routes on a schedule chosen to best serve the transit-riding public. I've never really understood any example involving a poisson process. They always seem to involve bus arrivals or light bulbs burning out, and I can't understand why the memory less property would ever make any sense for these. Even if the bus system was poorly run, why would it make sense to assume that the expected value of time to arrival doesn't change based on how long you've been waiting? What is an actual phenomenon that is well modeled by a poisson process?
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