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The Waiting Time Paradox, Or, Why Is My Bus Always Late?

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Re: The Waiting Time Paradox, Or, Why Is My Bus Always Late?

#71

Post by ikeboy »

8note wrote: post #66
ikeboy wrote: post #39

Earlier quoted context omitted.

Parent is talking about bitcoin, where that is false. If they are assuming the average, then they're assuming something false.

the important thing is that it doesn't matter what the parent assumed. whether the actual time is 10 minutes or 100 years, knowing that somebody else solved one recently doesn't speed up your time to find one

Of course it doesn't speed up your own time, since you have perfect information about your own hashpower. But it does tell you information about the total hashpower that's online, statistically.

I'll give an extreme example to make this clearer. Suppose 10X hashpower just came online an hour ago. It's quite likely that ~60 blocks have been found in the last hour, assuming the difficulty adjustment hasn't happened since. Seeing this, one could deduce that hashpower went up by ~10 and that the expected time till next block is roughly 1 minute instead of 10.

Now, in most cases hashpower doesn't change that drastically but it remains true that recent block times give you more than 0 information about hashpower and therefore about the expectation for future block times.

Re: The Waiting Time Paradox, Or, Why Is My Bus Always Late?

#72

Post by Pristina »

DalekBaldwin wrote: post #62
repsilat wrote: post #59

Earlier quoted context omitted.

This reminds me of a mathematical paradox that makes me doubt your conclusion: "In this country, every couple wants to have one daughter. They keep having children until they have a daughter, and then they stop. What gender balance should we expect?" Couples can have any number of sons, and every couple has exactly one daughter. Still, the accepted mathematical solution is an equal gender ratio for the couples' child…

The expected number of sons is 1, and the expected number of daughters is 1 (by the framing of the problem, in every possible scenario, there is exactly one daughter), but the expected value of the ratio is not 1:1. E[X]/E[Y] = E[X/Y] is not a valid identity. http://www.thebigquestions.com/2010/12/22/a-big-answer-2/

I read that and it seems wrong. The question asked "what fraction of the pop is female" but his argument is that 3 families of 4 girls and 1 family of 12 boys make the fraction of girls in the average family 75% (the average of 100% x3 and 0% x1) which is non-sensical to me.

>E[X]/E[Y] = E[X/Y] is not a valid identity.

is completely irrelevant here because it is being used to point out that a non-answer is wrong.

Re: The Waiting Time Paradox, Or, Why Is My Bus Always Late?

#73

Post by FabHK »

ikken wrote: post #26

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 man…

And a related counterintuitive fact (again, assuming 10 minutes):

1. If you pick a block randomly (uniformly), its average length is 10 minutes.

2. If you pick a point t0 in time randomly (uniformly), the average length of the block you're in is 20 mins (and the average length from t0 to next block is 10 mins, and the average length from previous block to t0 is also 10 mins (and, needless to say, 10+10=20...)).

Re: The Waiting Time Paradox, Or, Why Is My Bus Always Late?

#74

Post by adrianN »

asdkhadsj wrote: post #19
herodotus wrote: post #8

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…

Wow, interesting idea! Imagine if (in the future) some item like a phone can detect this information around you, and automatically record it. Forming games ontop of this life data would be weird, neat, fun and sad all at the same time. Imagine seeing a real example of where someone else is just more lucky than you are in stupid but impactful (on your morale) ways. If it didn't seem so tedious to track, I'd love to im…

Similar to the recruiters that throw away the top half of the application stack because they don't want unlucky people in their company I could see such data become valuable to some people.

Re: The Waiting Time Paradox, Or, Why Is My Bus Always Late?

#76

Post by jobigoud »

I've encountered the inspection paradox in debates about factory farming and people talking past each other points.

If you take the average farm, chances are that it's doing humane farming. But if you take the average animal, it has an overwhelming chance of being in an industrial farm.

Re: The Waiting Time Paradox, Or, Why Is My Bus Always Late?

#77

Post by anotheryou »

kuu wrote: post #70

A bit off-topic: How can you integrate a jupyter notebook in a blog post like this one? It looks really nice! Nice article, btw, interesting topic!

if I'm not mistaken there is a html export function that bakes it in to a static html

Re: The Waiting Time Paradox, Or, Why Is My Bus Always Late?

#78

Post by jobigoud »

twtw wrote: post #13

> 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.…

> 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?

I don't think it's saying anything about how long you've been waiting, and you don't know when was the last arrival.

It's saying that if you pick a random point on the timeline, the expected wait time doesn't change. That's because by taking a random point you have more chances of landing in a larger stretch of wait time than in a smaller one.

Re: The Waiting Time Paradox, Or, Why Is My Bus Always Late?

#79

Post by jobigoud »

ChrisFoster wrote: post #47

It strikes me that even with a perfectly regular starting schedule, buses might clump together in time because the schedule is probably dynamically unstable. To explain, picking up passengers from a stop costs time and a long time between buses implies a high probability that passengers will be waiting at a given stop. This further adding to the delay and shortens the time to the next bus in the schedule. I'm sure dr…

I think another confusing factor about that specific example is that bus shouldn't ever start before their schedule. Otherwise you run the risk of a bunch of people missing their bus even though they showed up on time. I think bus, trains and planes can only be late.

For example this is an article about a Japanese Train company issuing a public apology for departing 20 seconds early. https://www.bbc.com/news/world-asia-42009839

Re: The Waiting Time Paradox, Or, Why Is My Bus Always Late?

#80

Post by pwagland »

jon_richards wrote: post #69
Terr_ wrote: post #64

Earlier quoted context omitted.

I think the "paradox" comes from how people implicitly assume "any number of sons" is somehow distributed or weighted in a way that favors towards numbers of 1 or above. In contrast, "0 sons" is going to describe a full half of all marriages.

Same situation with the bus.

Not really. In the son/daughter case, the calculations are: expected daughters: 1 expected sons: 1/20 + 1/41 + 1/82 + 1/163 + 1/324 + …

So number of expected daughters = 1, number of expected sons = 1. In practice since women can't have an infinite number of children, then this wouldn't be an infinite series, so the real number of expected boys would be lower than one, but there you go…

Now, for the bus case, you get +1 if your bus turns up first, and -1 for every other bus that turns up first. Assume that it is completely random, then: expected + score is: 1/2 1 expected - score is: 1/2 * -1 + 1/4 * -2 + …

Expected + is 0.5, expected - is -1.

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