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I've decided that in order to keep my brain from rotting - it can happen quickly in business if you're not careful - I thought I'd embark on a project to itemize and deepen my understanding of probability. So here we go:
Below are some study notes on probability - more for personal reference than anything else (I tend to lose notebooks). This'll also give me a good excuse to work my LaTex skills.
**Sample Space & Events: **
The set of all possible experimental outcomes is called the sample space, S. An event, A, is a set of outcomes - a subset of S. Note that $emptyset $ and S itself are subsets of S, and hence are considered events. Events can be combined to form new events using set operators: $ A cap B $ is the event that occurs iff A OR B occurs (or both) $ A cup B $ is the event that occurs iff A AND B occurs $ A^complement$ is the complement of A - the event that occurs if A does NOT occur. Two events are mutually exclusive if $A cap B = emptyset $
Finite Probability Spaces - **(Equiprobable space): **
Suppose S contains n points in an equiprobable space - the various outcomes of the sample space have equal probabilities - then the probability of each point is 1/n. Note this cannot be generalized to spaces that are non-equiprobable. If an event A contains r points then the probability is r(1/n) = r/n We can say that $$P(A) = frac { n(A)} {n(S)} $$, where n(A) is the number of elements in a set A. The saying, "at random" can only apply to an equiprobable space.
Finite Probability Spaces - (Addition Principle): For any events A and B, $$ P(Acup B) = P(A) + P(B) - P(A cap B) $$
Conditional Probability:
Suppose E is an event in S with P(E) > 0. The probability that an event A occurs once E has occurred - *the conditional probability of A given E - *is given by, $$ P(A|E) = frac {P(A cap E)} {P(E)} $$. If S is an equiprobable space with events A and E, then, $$ P(A|E) = frac {n(A cap E)} {n(E)} $$
**(multiplication theorem for conditional probability): **
If we take the definition of conditional probability, $$ P(B|A) = frac {P(A cap B)} {P(A)} $$ and multiply the both sides by P(A), we get the following useful definition: $$ P(A cap B) = P(A) P(B|A) $$ Note this can easily be extended to three or more events.
Independent Events:
Events A and B are independent if $$P(Acap B) = P(A) P(B)$$, otherwise they are dependent.
**Independent Repeated Trials, Binomial Distribution...**Repeated trials with two outcomes, Bernoulli Trials, Binomial Experiment:
p = probability of success; q = 1 - p, the probability of failure. A binomial experiement consists of a fixed number of Bernoulli trials with B(n, p) denoting a binomial experiment with n trials with p probability of success. The probability of k successes, P(k) = $$ C_r = {nchoose k} = frac {n!}{k!(n-k)!} $$ $$ [P(E) = {n choose k} p^k (1-p)^{ n-k} $$
Random Variables:
We frequently wish to assign a number to each outcome in an experiment. For example, if we flip a coin, the outcomes are H or T; if we toss a pair of dice the outcomes are a pair of integers. Let's say we want to assign 1 to H, or assign the sum of the two integers to the outcome. For this we will use a random variable: A random variable $mathrm{X}$ is a rule that assigns a numerical value to each outcome in a sample space S. Let $R_mathrm{X}$ denote the set of numbers assigned by a random variable $mathrm{X}$. $R_mathrm{X}$ is called the range space.
TBC in a later post....