{"id":802,"date":"2014-03-10T20:15:22","date_gmt":"2014-03-11T00:15:22","guid":{"rendered":"http:\/\/learninglover.com\/blog\/?p=802"},"modified":"2014-07-03T13:07:30","modified_gmt":"2014-07-03T17:07:30","slug":"probability-sample-spaces","status":"publish","type":"post","link":"https:\/\/learninglover.com\/blog\/index.php\/2014\/03\/10\/probability-sample-spaces\/","title":{"rendered":"Probability: Sample Spaces"},"content":{"rendered":"

I’ve been doing a few games lately (can be seen here<\/a>, here<\/a> and here<\/a>) and, while I think those are very good ways to become interested in some of the avenues of math research, I also have had a few people come to me with questions regarding help with their classes. So I decided to write a script to try to help understand some elementary probability theory<\/a>, focusing on discrete sample spaces. <\/p>\n

\"Probability<\/p>\n

In statistics, any process of observation is referred to as an experiment.
\nThe set of all possible outcomes of an experiment is called the sample space<\/strong> and it is usually denoted by S. Each outcome in a sample space is called an element<\/strong> of the sample space. An event<\/strong> is a subset of the sample space or which the event occurs. Two events are said to be mutually exclusive<\/strong> if they have no elements in common.<\/p>\n

Similar to set theory, we can form new events by performing operations like unions, intersections and compliments on other events. If A and B are any two subsets of a sample space S, then their union A ∪ B is the subset of S that contains all the elements that are in either A, in B, or in both; their intersection A ∩ B is the subset of S that contains all the elements that are in both A and B; the compliment<\/strong> A’ of A is the subset of S that contains all the elements of S that are not in A.<\/p>\n

A probability is a function that assigns real numbers to events of a sample space. The following are the axioms of probability that apply when the sample space is discrete (finite or countable).<\/p>\n

Axiom 1: The probability of an event is a non-negative real number; that is P(A) ≥ 0 for any subset A of S.
\nAxiom 2: The probability of the entire sample space is 1; that is P(S) = 1.
\nAxiom 3: If A1, A2, A3, … , is a finite or infinite sequence of mutually exclusive events of S, then
\nP(A1 ∪ A2 ∪ A3 ∪ …) = P(A1) + P(A2) + P(A3) + …
\nIf A and B are any two events in a sample space S and P(A) ≠ 0, the conditional probability of B given A is <\/p>\n\n\n
P(B | A) = <\/td>\n\n\n\n
P(A ∩ B)<\/p>\n
\n

P(A)<\/td>\n<\/tr>\n<\/table>\n<\/td>\n<\/tr>\n<\/table>\n

Two events A and B are independent if and only if P(A | B) = P(A) ∙ P(B).<\/p>\n","protected":false},"excerpt":{"rendered":"

I’ve been doing a few games lately (can be seen here, here and here) and, while I think those are very good ways to become interested in some of the avenues of math research, I also have had a few people come to me with questions regarding help with their classes. So I decided to […]<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[],"tags":[],"class_list":["post-802","post","type-post","status-publish","format-standard","hentry"],"_links":{"self":[{"href":"https:\/\/learninglover.com\/blog\/index.php\/wp-json\/wp\/v2\/posts\/802","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/learninglover.com\/blog\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/learninglover.com\/blog\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/learninglover.com\/blog\/index.php\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/learninglover.com\/blog\/index.php\/wp-json\/wp\/v2\/comments?post=802"}],"version-history":[{"count":1,"href":"https:\/\/learninglover.com\/blog\/index.php\/wp-json\/wp\/v2\/posts\/802\/revisions"}],"predecessor-version":[{"id":894,"href":"https:\/\/learninglover.com\/blog\/index.php\/wp-json\/wp\/v2\/posts\/802\/revisions\/894"}],"wp:attachment":[{"href":"https:\/\/learninglover.com\/blog\/index.php\/wp-json\/wp\/v2\/media?parent=802"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/learninglover.com\/blog\/index.php\/wp-json\/wp\/v2\/categories?post=802"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/learninglover.com\/blog\/index.php\/wp-json\/wp\/v2\/tags?post=802"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}