Is the inspection paradox what would happen if you surveyed everyone on how many siblings they had, and every sibling double-counted N-1 times (where N is the number of siblings in their family), inflating the resulting "average number of siblings", or is that something different?
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?
#62Earlier quoted context omitted.
Because we might score -1, -2 or worse if 2 or 3 busses went in the other direction before ours came, but if ours came first, we score 1. We get on the bus and thus don’t know if another one or more arrives first on our side.
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.
Re: The Waiting Time Paradox, Or, Why Is My Bus Always Late?
#63> 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.…
>What is an actual phenomenon that is well modeled by a poisson process?
I used it in a cell tissue simulation where the user could define how frequently the cells divide. If you start with 100 cells and want them each to divide, on average, every N iterations, using a Poisson formula to decide if a cell splits or not based on a random number is ideal, very precise (in the aggregate), and avoids a lot of odd artifacts.
Re: The Waiting Time Paradox, Or, Why Is My Bus Always Late?
#64Earlier quoted context omitted.
Because we might score -1, -2 or worse if 2 or 3 busses went in the other direction before ours came, but if ours came first, we score 1. We get on the bus and thus don’t know if another one or more arrives first on our side.
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…
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.
Re: The Waiting Time Paradox, Or, Why Is My Bus Always Late?
#65Is the inspection paradox what would happen if you surveyed everyone on how many siblings they had, and every sibling double-counted N-1 times (where N is the number of siblings in their family), inflating the resulting "average number of siblings", or is that something different?
This is my first exposure to it, but yes, I think so. My paraphrase is: The Inspection Paradox is when you ask someone "in the mix" about the mix. You're only going to get an accurate estimation of the mix by standing outside of the mix.
So yes, if you want an accurate count of siblings, you would consult some spreadsheet that just lists how many children each family had. If you go and start asking the families themselves (those "in the mix") then your results will be skewed.
I thought the article that this article linked to was also very good, "The Inspection Paradox Is Everywhere," by Allen Downey, http://allendowney.blogspot.com/2015/08/the-inspection-parad...
Re: The Waiting Time Paradox, Or, Why Is My Bus Always Late?
#66Earlier quoted context omitted.
What you say is not a rebuttal of the parent comment. Parent explicitly said that average block times are 10mins in the assumption. The most recent block time doesn't change that.
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
Re: The Waiting Time Paradox, Or, Why Is My Bus Always Late?
#67Hah great analysis. One factor with bus' is the schedule is likely planned to minimize early arrivals at the risk of being late more often. Usually when a bus is early it has to sit and wait until its departure time. A late running bus can be more efficient, and if kept until departure time might not ever get a chance to average down the bursts of lateness.
a very late bus grabs everyone from each stop, so the next bus ends up early because it skips a bunch of less popular stops
Re: The Waiting Time Paradox, Or, Why Is My Bus Always Late?
#68I’m glad I live in east Asia. Busses and trains are almost never late
Re: The Waiting Time Paradox, Or, Why Is My Bus Always Late?
#69Earlier 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…
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.
Re: The Waiting Time Paradox, Or, Why Is My Bus Always Late?
#70A 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!