Well, here it is. A month where I'm supposed to put up a new blog post each day... why did I choose a month with so many days again? Since I will be producing prose at such a prodigious rate, I cannot guarantee, in fact I don't even suspect, that what I write will be up to what I consider to be my normal standard of interestingness and thought-provoketivity. Take this post for example.
In fact, I still have a little over a week left in the semester, so for that time I shall be scrambling to post something each day. However, I promise not to fall back to updates about my life too often; I get bored writing those and I can only imagine how bored you must be reading them. One of my favorite things about this blog is how much enjoyment I get out of going back and re-reading things I have posted. When I was searching for my favorite posts to list in my 100th post I saw a lot of interesting things that I had forgotten that I'd written. Of course, my absolute favorite thing about this blog is the discussions that will occasionally pop up in the comments, but looking back and realizing how clever I am is up there.
After school gets out I should have a bit more time to put into my posts. There are various topics that I have wanted to talk about, but have yet to find the time and or motivation to put together the polished post that I think they deserve. Maybe I'll crack open my cache of blog ideas and get some use out of that. Then I head out for adventure! Or, maybe just on vacation to visit family and friends. Either way, I'll be busier, so maybe back to the inane, "this is what I am doing," type posts, but at least they will be slightly more interesting.
I did not write up a bunch of posts so I have one on standby in case I get too busy to write one on a given day, both because this doesn't seem in the spirit of NaNoWriMo, and because it seems ridiculous for me to stop publishing in my blog to save up for a plan that is supposed to get me to publish in my blog more. However, now that December has begun I reserve the right to write posts ahead of time and publishing them as needed to keep on schedule. Doing something that prepared doesn't really sound like something I would do, but stranger things have happened. I'll see you tomorrow!
Thursday, December 1, 2011
Sunday, November 13, 2011
Bully For You
I have been hearing about Michigan's Anti-Bullying bill from many people, and now that multiple Oregonian friends have contacted me regarding it, I felt like it might be good for me to clarify my position regarding the bill. This was not a simply thing, because while the legal issues are simple enough, the surrounding cultural issues, which are really the issues here, are quite complex. First, here is the text of the bill as passed by the senate and here is the article that I think best explains the situation.
I am not sure why the exemption section was written in, because it doesn't do anything. I am not sure why people are reacting in shock or horror, because it doesn't do anything. I am not sure why Senator Whitmer felt the need to go Senator McCarthy on us. As far as I can tell, even if the exemption did something, it would not permit the type of bullying experienced by the student for whom the law is named. Because, although the student happened to be gay, it appears that the bullying occurred as part of a hazing, with no relation to his sexual orientation or the tormentors religious beliefs or moral convictions. I'm not sure why we think we can legislate civility, and I think it exposes a fundamental flaw in our society that we should try.
If I had to guess what reasoning led to the exemption I would guess the following. I doubt that they wanted to protect "religious" students who proceed to beat the stuffing out of a gay student, or an atheist, or a Christian of another denomination. In fact, despite rhetoric from opponents indicating that the law does this, section 8 specifies that it protects, "a STATEMENT of a sincerely held religious belief or moral conviction," (emphasis mine). Note that this does not apply to calling people of any group by a derogatory label, using the word "gay" as a derogatory euphemism, or holding an atheist's head in a toilet (unless one interprets "statement" in an incredibly lose sense). Rather, I would guess that some people are feeling defensive after losses experienced in areas such as voluntary school prayer and freedom to perform religious works in school and are seeking to prevent this legislation from being used to suppress statements such as, "due to your lifestyle you are going to Hell." While I do not feel that such statements are civil, or even Biblicaly sound, I do not believe that the should, in and of themselves, be prohibited. Should teachers be permitted to say them to students in a specifically educational setting, probably not, but I don't think that students need to be kept from saying that.
Understanding the shock and horror of the response is much easier. Many people have been exposed to the rhetoric that this will protect gay bashers, in the euphemistic or even literal usage, as long as they utter a perfunctory, "God's will be done," before they lay into their victim. If I thought that this bill did that I would be shocked and horrified too. So, let us focus on why people who probably ought to know better, such as our legislators, are appalled by this bill. Part of it might be that our political system doesn't necessarily tend to produce politicians with more ability to understand law or motivation to try than the average journalist. But another factor is that it does kind of, sort of, look like a religious group is trying to codify their persecution of another group as legally protected. But that perception stems from a persistent failure of churches to deal with the public perception that they are bastions of violence against homosexuals, which is not entirely relevant when writing legislation.
Underlying all of this has been my belief that our lives and the realm of legalities are two very separate things. Supporting this belief is my having lived my entire life without directly interacting with the courts nor being able to recall a single instance where fear of legal repercussions caused me to do something I didn't want to. While the law fashions an overarching penumbra on our society, I do not think the two directly interact too often. As such, I would encourage the consideration of cultural solutions to problems of incivility. Teachers have a job, and that is to educate their students. While they should model appropriate behavior to their students, I do not think the current education system is structured in such a way that they ought to be responsible for instilling such behavior in their students. Specifically, the low teacher to student ratio and the lack of institutional support for teachers attempting to discipline their students lead me to believe that teachers cannot adequately identify or respond to all the social problems that their students may exhibit.
The decline of a stay at home parent also seems problematic since this reduces the opportunities that a child has to interact with an adults in a smaller child to adults ratio where undesirable behaviors can be better identified and addressed at greater length. While there are regrettable economic realities behind this decline, I also think there are social improvements that have contributed to it. As such, I do not feel that simply rolling back the legal gains that divorce has made is at all a satisfactory solution. Rather, let us examine other societal changes we might effect. Reaffirming our nation's commitment to marriage might be one of them, and I don't mean trying to keep same sex couples from having it, or forcing battered women to endure it, but examining why so many frivolous marriages occur. Improving our access to contraception might be another, having a child unintentionally does not bode well for one's preparedness for parenthood. Examining the nature of our communal bonds might also be worthwhile, as it may not necessarily be a child's own parents who need to provide their adult supervision, as the saying "it takes a village to raise a child," indicates.
Finally, we need adults who are more exposed to the variety of lifestyles that exist in our nation. "Advances" in transportation and telecommunication have actually precipitated declines in human interaction. Because we CAN communicate with more people we tend to actually communicate with with people who are more like us, as we have a wider pool from which to find them. To offset this I advocate for the dual responses of more integrated communities and more local, on the neighborhood level, government and social groups. Of course, these solutions are entirely inappropriate to enforce through legal methods, but that doesn't mean that they are not the solutions we need for the problems that we are facing.
Note, this does not express a dislike for the bill in either of its forms, as all the bill does is require schools put policies into place to deal with bullying insofar as it is detrimental to the educational environment.
Live Your Life
For over a week now I have had a draft of my next post on Identity open, waiting for me to straighten out my thoughts enough to forge ahead on it. However, I tend to think dialogically, so presenting my thoughts in a narrative format can be difficult. So, in the mean time let me post something less substancial.
Toward the beginning of this calendar year, when I was updating at a good rate, I ran a series of thoughts inspired by pop music. This song did not make the series because it doesn't really inspire interesting academic type thoughts, but I love the message of accepting that the past has happened, then dealing with being one's self in the present. I'm not happy with the slow rate that I am posting here, but I am living my life as best I can, and I stand by that!
Tuesday, November 1, 2011
MoBloYoFoMo
So, I haven't been writing in my blog as much as I would like. Not only because, if responses are an accurate judge, interest is down, but also because I have ideas bouncing around in my head I would like to have feed back upon. I considered resolving to blog every day in November, as sort of a tribute to National Novel Writing Month (NaNoWriMo), but I'm really freakin' busy. Also, a couple of my readers are participating in NaNoWriMo, and it seems poor planning to produce tons of my own content that I want them to read as they are also trying to write a book in a month.
So, December will be the first ever More Blogging You Fool Month, or MoBloYoFoMo. And if you hear that spoken in the voice of a Judoon, you are not wrong. Basically, I will be updating my blog every day next month! So, I hope you tune in to see what happens!
So, December will be the first ever More Blogging You Fool Month, or MoBloYoFoMo. And if you hear that spoken in the voice of a Judoon, you are not wrong. Basically, I will be updating my blog every day next month! So, I hope you tune in to see what happens!
Monday, October 17, 2011
Identity
Identity, it's who we are. But it isn't just who we are to ourselves, it is also who we are to other people. I shall try to keep my thoughts reasonably linear, but this is a topic that has come up in a variety of facets recently, so it may be difficult to isolate it from the surrounding context to discuss in a stand alone fashion.
One important facet of our identity is our cultural heritage. This, in some sense, grounds us to a specific place or places in history, "our ancestors were these people and came from here." It might be argued that many people in the United States lack this sense of heritage, but I think that it would be more accurate to say that they have incorporated the American ideal of settlers and intermixing as their heritage.
The question arises, what must be done to lay claim to a cultural heritage? I would assert that the act of laying honest claim to such a heritage validates itself. Of course, if I were to jokingly claim to be heir to the royal traditions of the Incas, this would not carry a lot of weight. However, if someone genuinely believes themselves to be an inheritor of a specific culture, then it is self evident that their identity is shaped by a sense of inclusion in that culture.
On the other hand, one might hold that, in order to possess a cultural heritage, one must take an active role in exploring that culture. While this is a perfectly valid way to organize people into culture groups, I think it is unnecessarily restrictive. At the risk of seeming ego maniacal, I will provide an illustrative example from my own life, simply because it is the only life with which I am familiar enough to say things with confidence.
One important facet of our identity is our cultural heritage. This, in some sense, grounds us to a specific place or places in history, "our ancestors were these people and came from here." It might be argued that many people in the United States lack this sense of heritage, but I think that it would be more accurate to say that they have incorporated the American ideal of settlers and intermixing as their heritage.
The question arises, what must be done to lay claim to a cultural heritage? I would assert that the act of laying honest claim to such a heritage validates itself. Of course, if I were to jokingly claim to be heir to the royal traditions of the Incas, this would not carry a lot of weight. However, if someone genuinely believes themselves to be an inheritor of a specific culture, then it is self evident that their identity is shaped by a sense of inclusion in that culture.
On the other hand, one might hold that, in order to possess a cultural heritage, one must take an active role in exploring that culture. While this is a perfectly valid way to organize people into culture groups, I think it is unnecessarily restrictive. At the risk of seeming ego maniacal, I will provide an illustrative example from my own life, simply because it is the only life with which I am familiar enough to say things with confidence.
Among my mix of European ancestors, I have some Germans and some Italians. And, while this has led me to study a little bit of German and put some effort into learning about German culture, I identify more strongly as Italian, despite having invested less effort in my Italian identity. I believe this is simply because my last name, if you unravel some Ellis Island mishaps, is Italian. So, just by identifying myself by name I am asserting my Italian identity, and it has caused people to react to me differently, thus further reinforcing my "Italian-ness."
And here we see how one's self identity and the manner in which others construct identity groups have such tricky interrelation. As I mentioned earlier, I think organizing people into identity groups based on putting effort into claiming a cultural identity is a valid way of organizing people, by which I mean that the groups that one obtains from such organization will likely share some distinguishing characteristics of interest and this organization doesn't seem to inherently promote ethnic cleansings. However, it may be quite at odds with how other people think about themselves, and indeed, how other people organize the world into cultural groups. So, it seems important to acknowledge that people can come to their identity through a variety of methods.
Further confusing the matter is the practice of assuming identities. For example, a European manga/anime enthusiast who feels their hobby confers a cultural legacy from Japan. First, let me note that this does not apply to every manga/anime enthusiast, just those who feel their hobby includes them in the penumbra of Japanese culture. Secondly, while this notion of cultural heritage may be at odds with the traditional notion of culture as inheritable from one's parents, it is entirely in keeping with both my criterion and, it seems to me, with the contrasting criterion from above, as the person in question both self identifies with Japanese culture and is investing effort into the cultivation of this cultural identity.
This highlights a further complication, in that essentializing Japanese culture down to anime and manga; or even anime, manga, and quirky gadgets; does Japanese culture a disservice, and, depending on one's views, may even be insulting. However, otaku are a part of Japanese culture, so one perhaps should not say that this view of Japanese culture is incorrect, insomuch as it is incomplete. So, toward what concept of a culture ought we direct our efforts in order to "earn" cultural inclusion?
So far we have mentioned three reasonable but incompatible methods for assigning cultural heritage; self identification, cultural participation, and parental inheritance. These are by no means exhaustive, geographical inheritance (Italians are people who have lived in Italy for some duration, which varies from person to person) and nationalism (Italians are citizens of Italy), for example, also have adherents. In light of this complexity, I am inclined to broaden my original position. While I still believe in accepting how others identify themselves, I also believe that we should accept that other people will have other ways of organizing people into cultural groups. That said, not all such organizations are "reasonable," which I would like to address at a later date.
And here we see how one's self identity and the manner in which others construct identity groups have such tricky interrelation. As I mentioned earlier, I think organizing people into identity groups based on putting effort into claiming a cultural identity is a valid way of organizing people, by which I mean that the groups that one obtains from such organization will likely share some distinguishing characteristics of interest and this organization doesn't seem to inherently promote ethnic cleansings. However, it may be quite at odds with how other people think about themselves, and indeed, how other people organize the world into cultural groups. So, it seems important to acknowledge that people can come to their identity through a variety of methods.
Further confusing the matter is the practice of assuming identities. For example, a European manga/anime enthusiast who feels their hobby confers a cultural legacy from Japan. First, let me note that this does not apply to every manga/anime enthusiast, just those who feel their hobby includes them in the penumbra of Japanese culture. Secondly, while this notion of cultural heritage may be at odds with the traditional notion of culture as inheritable from one's parents, it is entirely in keeping with both my criterion and, it seems to me, with the contrasting criterion from above, as the person in question both self identifies with Japanese culture and is investing effort into the cultivation of this cultural identity.
This highlights a further complication, in that essentializing Japanese culture down to anime and manga; or even anime, manga, and quirky gadgets; does Japanese culture a disservice, and, depending on one's views, may even be insulting. However, otaku are a part of Japanese culture, so one perhaps should not say that this view of Japanese culture is incorrect, insomuch as it is incomplete. So, toward what concept of a culture ought we direct our efforts in order to "earn" cultural inclusion?
So far we have mentioned three reasonable but incompatible methods for assigning cultural heritage; self identification, cultural participation, and parental inheritance. These are by no means exhaustive, geographical inheritance (Italians are people who have lived in Italy for some duration, which varies from person to person) and nationalism (Italians are citizens of Italy), for example, also have adherents. In light of this complexity, I am inclined to broaden my original position. While I still believe in accepting how others identify themselves, I also believe that we should accept that other people will have other ways of organizing people into cultural groups. That said, not all such organizations are "reasonable," which I would like to address at a later date.
Friday, October 7, 2011
Occupy the Middle!
Before I grow fully involved in talking about identity groups, I would like to finally hash out this post about Occupy Wall Street that I have been trying to write for over a week. Despite, or maybe because of, nebulous and conflicting concerns and desires, the Occupy Wall Street movement has inspired a plethora of related movements, occupying things across the United States and even internationally.
Ideally I might open with what the goals of the movement are. However, as I have noted, such a thing is not easy to formulate. I think it is safe to say that the movements are characterized by an anti-capitalist sentiment, and an accompanying dissatisfaction with the growing economic disparity within the US. The proposed solutions are, unfortunately, less homogeneous.
Ideally I might open with what the goals of the movement are. However, as I have noted, such a thing is not easy to formulate. I think it is safe to say that the movements are characterized by an anti-capitalist sentiment, and an accompanying dissatisfaction with the growing economic disparity within the US. The proposed solutions are, unfortunately, less homogeneous.
I think that this is a necessary characteristic of the movements. In order to enjoy such broad popular support, I don't think one can afford to commit to a more specific plan. Ironically, one can witness the same phenomenon within the very political structure that, many of, the protesters vilify. There, just as with the Occupy Wall Street movement, it is arduous to nail down exactly what a politician plans to do, simply because it is easier to garner support by pointing out what problems exist than by proposing specific measures by which one intends to correct them.
By no means to I consider this to be a scathing criticism of the Occupy Wall Street movement. I think it is inherently valuable to gain a platform from which we may air our grievances and question whether practices historically considered social necessities are, actually, contingencies put into place to tailor the system to the benefit of one group or another. However, I believe that true change will not be effective until people develop a plan of action more sophisticated than sitting in a public place until someone else gets sufficiently fed up and fixes the problem to make you go away.
While some have likened Occupy Wall Street to 60's era activism, the movement that most reminds me of Occupy Wall Street is the Tea Party. While they evolved out of radically different ideologies, I think at the heart of both movements is a deep sense of broad dissatisfaction with current domestic affairs. Contrast this with peace protests, civil rights movements, and even religious demonstrations, which all have much more definite goals to which they aspire.
Of course, there are significant differences between the Tea Party and the Occupy Wall Street movements. I think that the plans of the Tea Party were much more concrete, even if the underlying goals which they hoped to thereby accomplish remained quite nebulous. Perhaps correspondingly, I think the aims of the Tea Party translate better into the language of policy and legislation. Unfortunately, I do not particularly believe the change for which the Tea Party advocates can jar our society out of the malaise in which it currently lingers.
While I may not identify with"the 99%," I think their call for a re-conceptualization of "the good life" and our fundamental societal systems is a needed one. That said, I do not see how it is a matter of policy. One cannot legislate what people value, nor an ethic of individual accountability and responsibility. Of course, one might be able to, through legislation, encourage such a culture to develop. Personally I would like to see our nation take moral education and civic involvement more seriously, but that is just my opinion.
While some have likened Occupy Wall Street to 60's era activism, the movement that most reminds me of Occupy Wall Street is the Tea Party. While they evolved out of radically different ideologies, I think at the heart of both movements is a deep sense of broad dissatisfaction with current domestic affairs. Contrast this with peace protests, civil rights movements, and even religious demonstrations, which all have much more definite goals to which they aspire.
Of course, there are significant differences between the Tea Party and the Occupy Wall Street movements. I think that the plans of the Tea Party were much more concrete, even if the underlying goals which they hoped to thereby accomplish remained quite nebulous. Perhaps correspondingly, I think the aims of the Tea Party translate better into the language of policy and legislation. Unfortunately, I do not particularly believe the change for which the Tea Party advocates can jar our society out of the malaise in which it currently lingers.
While I may not identify with"the 99%," I think their call for a re-conceptualization of "the good life" and our fundamental societal systems is a needed one. That said, I do not see how it is a matter of policy. One cannot legislate what people value, nor an ethic of individual accountability and responsibility. Of course, one might be able to, through legislation, encourage such a culture to develop. Personally I would like to see our nation take moral education and civic involvement more seriously, but that is just my opinion.
Saturday, October 1, 2011
The Abstract Power of Mathematics
Yesterday, as I was walking home, I stepped upon an acorn just to hear the popping sound it makes. Stirred by a moment of pique, I picked it up to investigate what was inside. To my disgust, I saw a couple of maggot/grubby things, which made me feel bad about stepping on it, and all the other acorns upon which I have trod. This got me to wondering, how likely is it that a given acorn has a grub in it?
Since the first one I investigated had a grub in it, I got the impression that it would probably be a fairly likely thing. If only one in every hundred contained a grub, it would be terribly improbable that the first one I examined would just happen to have one. However, I wasn't interested enough to try to remember how conditional probability worked, so I just went home.
After I got home, I went to pay my rent. Walking out of the rental office I saw a maintenance van, which was apparently number 1 in their fleet. Since the van I saw was labelled one, I started wondering about the size of their fleet. If you see a van with a certain number, you can assume the fleet is at least that large, but it doesn't tell you how big it actually is. So, seeing a van with number one seems to indicate a small fleet, otherwise I would be much more likely to see a van with a large number. When I realized that this was, basically, the same problem that had occurred to me earlier, with the grubs in acorns, I decided that who was I to ignore coincidence, and decided to work out the conditional probabilities.
Most of the time people think about probability knowing, approximately, how likely outcomes are and they want to figure out what will happen. For example, if you roll a six sided die, you would be interested in knowing what number you roll. Conditional probability works in reverse, if you know you are in one of a few scenarios and you know what happens, you try to figure out what the likelihood is that you are in a specific scenario. For example, if someone rolls a six sided, eight sided, or twenty sided die and they tell you that they roll a 7, you know they didn't roll the six sided die. For more information on determining conditional probabilities see below*
For simplicity, I decided to assume one out of every so many acorns has grubs. This ignores a large number of cases, for example 2 out of every 3 have them, but we can round those to the nearest considered case. I made this assumption both because it simplifies the calculations, and because it makes the acorn problem: the first acorn I see has grubs when 1 out of n do, the same as the van problem: the van I see is number 1 out of a fleet of n total vans.
Most of the time people think about probability knowing, approximately, how likely outcomes are and they want to figure out what will happen. For example, if you roll a six sided die, you would be interested in knowing what number you roll. Conditional probability works in reverse, if you know you are in one of a few scenarios and you know what happens, you try to figure out what the likelihood is that you are in a specific scenario. For example, if someone rolls a six sided, eight sided, or twenty sided die and they tell you that they roll a 7, you know they didn't roll the six sided die. For more information on determining conditional probabilities see below*
For simplicity, I decided to assume one out of every so many acorns has grubs. This ignores a large number of cases, for example 2 out of every 3 have them, but we can round those to the nearest considered case. I made this assumption both because it simplifies the calculations, and because it makes the acorn problem: the first acorn I see has grubs when 1 out of n do, the same as the van problem: the van I see is number 1 out of a fleet of n total vans.
One would sort of expect lower numbers in the fleet or higher grub frequency to be the most likely scenario. But, it turns out, if you consider fleets of any number of vans to be possible, then even a fleet of one van, the most likely, turns out to be statistically impossible, or a 0 probability event, if you remember my first post on probability. This is because, although the van's number is 1 100% of the time, or 1/1, if the fleet size is one, the probability of the van's number being one over all possible scenarios turns into the infinite sum of 1/1+1/2+1/3+1/4... which diverges**
Anyway, I suppose the main point of this post is not conditional probability, or the odd things that happen when you let infinity enter things, but rather how wonderful it is that math can take two, ostensibly different problems, like vans in a fleet and grubs in an acorn, and unify them into one, underlying concept.
*Calca I: Conditional Probability
Suppose that you know something happened, say someone rolled a 7 on a die, but you don't know exactly what circumstances led to this occurring, they either rolled a die with 8 sides or a die with 10 sides. To figure out how likely a scenario is given a known outcome, one simply divides the likelihood of the known outcome in that given scenario by the sum of the likelihoods of that outcome in all possible scenario.
For example, the 7 will roll 1/8 of the time on an 8 sided die and 1/10 of the time on a 10 sided die, so the likelihood that an 8 sided die was rolled to obtain the 7 is: (1/8)/(1/8+1/10), or 5/9. This makes a certain amount of sense, the 8 sided die is more likely to produce a 7 than the 10 sided die, so if a 7 rolled, we should think the 8 sided scenario more likely than the 10 sided one.
**Calca 2: The Series 1/n
If we try to find the probability that a fleet only has one van, given that we saw the van numbered 1, out of scenarios allowing the fleet to have any number of vans, we run into a problem. The numerator should be the probability of the observed event in the specified scenario, namely seeing van number 1 in a fleet of one van, or 1/1. However, the denominator should be the sum of the probability of seeing van number one over all scenarios, which is (1/1+1/2+1/3+1/4+...).
To get a feel for how to handle this sum, let us ignore the first term, 1/1. The next term, 1/2, contributes 1/2 towards the total. The next two terms, 1/3 and 1/4, are each greater than or equal to 1/4, and so together will contribute something at least 1/2 in size. The next four terms, 1/5, 1/6, 1/7, and 1/8, are each at least 1/8, so together they will contribute something at least 1/2 in size. Continuing on this way, we notice that the next 2^n terms are always at least 1/(2^(n+1)), and so together contribute something of at least size 1/2. Thus we can think of the sum as being larger than an infinite number of 1/2's being added together. This implies that the sum cannot be a finite number, so the conditional probability of the 1 van fleet scenario is 1 divided by a limit which approaches infinity, so the probability goes to zero.
Anyway, I suppose the main point of this post is not conditional probability, or the odd things that happen when you let infinity enter things, but rather how wonderful it is that math can take two, ostensibly different problems, like vans in a fleet and grubs in an acorn, and unify them into one, underlying concept.
*Calca I: Conditional Probability
Suppose that you know something happened, say someone rolled a 7 on a die, but you don't know exactly what circumstances led to this occurring, they either rolled a die with 8 sides or a die with 10 sides. To figure out how likely a scenario is given a known outcome, one simply divides the likelihood of the known outcome in that given scenario by the sum of the likelihoods of that outcome in all possible scenario.
For example, the 7 will roll 1/8 of the time on an 8 sided die and 1/10 of the time on a 10 sided die, so the likelihood that an 8 sided die was rolled to obtain the 7 is: (1/8)/(1/8+1/10), or 5/9. This makes a certain amount of sense, the 8 sided die is more likely to produce a 7 than the 10 sided die, so if a 7 rolled, we should think the 8 sided scenario more likely than the 10 sided one.
**Calca 2: The Series 1/n
If we try to find the probability that a fleet only has one van, given that we saw the van numbered 1, out of scenarios allowing the fleet to have any number of vans, we run into a problem. The numerator should be the probability of the observed event in the specified scenario, namely seeing van number 1 in a fleet of one van, or 1/1. However, the denominator should be the sum of the probability of seeing van number one over all scenarios, which is (1/1+1/2+1/3+1/4+...).
To get a feel for how to handle this sum, let us ignore the first term, 1/1. The next term, 1/2, contributes 1/2 towards the total. The next two terms, 1/3 and 1/4, are each greater than or equal to 1/4, and so together will contribute something at least 1/2 in size. The next four terms, 1/5, 1/6, 1/7, and 1/8, are each at least 1/8, so together they will contribute something at least 1/2 in size. Continuing on this way, we notice that the next 2^n terms are always at least 1/(2^(n+1)), and so together contribute something of at least size 1/2. Thus we can think of the sum as being larger than an infinite number of 1/2's being added together. This implies that the sum cannot be a finite number, so the conditional probability of the 1 van fleet scenario is 1 divided by a limit which approaches infinity, so the probability goes to zero.
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