{"id":2185,"date":"2013-06-19T17:09:06","date_gmt":"2013-06-19T21:09:06","guid":{"rendered":"http:\/\/scientopia.org\/blogs\/goodmath\/?p=2185"},"modified":"2013-06-19T17:09:06","modified_gmt":"2013-06-19T21:09:06","slug":"2185","status":"publish","type":"post","link":"http:\/\/www.goodmath.org\/blog\/2013\/06\/19\/2185\/","title":{"rendered":"Probability Spaces"},"content":{"rendered":"

Sorry for the slowness of the blog lately. I finally got myself back onto a semi-regular schedule when I posted about the Adria Richards affair, and that really blew up. The amount of vicious, hateful bile that showed up, both in comments (which I moderated) and in my email was truly astonishing. I’ve written things which pissed people off before, and I’ve gotten at least my fair share of hatemail. But nothing I’ve written before came close to preparing me for the kind of unbounded hatred that came in response to that post. <\/p>\n

I really needed some time away from the blog after that. <\/p>\n

Anyway, I’m back, and it’s time to get on with some discrete probability theory!<\/p>\n

I’ve already written a bit about interpretations<\/em> of probability. But I haven’t said anything about what probability means formally. When I say that the probability of rolling a 3 with a pair of fair six-sided dice is 1\/18, how do I know that? Where did that 1\/6th figure come from?<\/p>\n

The answer lies in something called a probability space<\/em>. I’m going to explain the probability space in frequentist terms, because I think that that’s easiest, but there is (of course) an equivalent Bayesian description.)\tSuppose I’m looking at a particular experiment. In classic mathematical form, a probability space consists of three components (Ω, E, P), where:<\/p>\n

    \n
  1. Ω, called the sample space<\/em>, is a set containing all possible outcomes of the experiment. For a pair of dice, Ω would be the set of all possible rolls: {(1,1), (1,2), (1,3), (1,4), (1,5), (1, 6), (2,1), …, (6, 5), (6,6)}.<\/li>\n
  2. E<\/em> is an equivalence relation over Ω, which partitions Ω into a set of events<\/em>. Each event is a set of outcomes that are equivalent. For rolling a pair of dice, an event is a total – each event is the set of outcomes that have the same total. For the event “3” (meaning a roll that totalled three), the set would be {(1, 2), (2, 1)}.<\/li>\n
  3. P<\/em> is a probability assignment<\/em>. For each event e<\/em> in E<\/em>, P(e)<\/em> is a value between 0 and 1, where:\n

    \t\t

    Sigma_{ein E} P(e) = 1<\/center><\/p>\n

    (That is, the sum of the probabilities of all of the possible events in the space is exactly 1.)<\/p>\n<\/li>\n<\/ol>\n

    The probability of an event e<\/em> being the outcome\tof a trial is P(e)<\/em>.<\/p>\n

    So the probability of any particular event as the result of a trial is a number between 0 and 1. What’s it mean? If the probability of event e<\/em> is p<\/em>, then if we repeat the trial N<\/em> times, we expect N*p<\/em> of those trials to have e<\/em> as their result. If the probability of e<\/em> is 1\/4, and we repeat the trial 100 times, we’d expect e<\/em> to be the result 25 times.<\/p>\n

    But in an important sense, that’s a cop-out. We’ve defined probability in terms of this abstract model, where the third component is the probability. Isn’t that circular?<\/p>\n

    Not really. For a given trial, we create the probability assignment by observation and\/or analysis. The important point is that this is really just a bare minimum starting point. What we really care about in probability isn’t the change associated with a single, simple, atomic event. What we want to do is take the probability associated with a group of single events, and use our understanding of that to allow us to explore a complex event. <\/p>\n

    If I give you a well-shuffled deck of cards, it’s easy to show that the odds of drawing the 3 of diamonds is 1\/52. What we want to do with probability is things like ask: What are the odds of being dealt a flush in a poker hand?<\/p>\n

    The construction of a probability space gives us a well-defined platform to use for building probabilistic models of more interesting things. Give a probability space of two single dice, we can combine them together to create the probability space of the two dice rolled together. Given the probability space of a pair of dice, we can construct the probability space of a game of craps. And so on.<\/p>\n","protected":false},"excerpt":{"rendered":"

    Sorry for the slowness of the blog lately. I finally got myself back onto a semi-regular schedule when I posted about the Adria Richards affair, and that really blew up. The amount of vicious, hateful bile that showed up, both in comments (which I moderated) and in my email was truly astonishing. I’ve written things […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":true,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2}},"categories":[53],"tags":[],"class_list":["post-2185","post","type-post","status-publish","format-standard","hentry","category-probability"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/s4lzZS-2185","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/2185","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/comments?post=2185"}],"version-history":[{"count":0,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/2185\/revisions"}],"wp:attachment":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/media?parent=2185"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/categories?post=2185"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/tags?post=2185"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}