POE Probability

union (or) of a &b

entire venn diagram

what is probability

the calculated likelihood that a given event will occur

intersection (AND) of a & b

middle part of venn diagram

bernoulli trial

A random experiment with exactly two possible outcomes: "success" and "failure", in which the probability of success is the same for each trial of the experiment. Trial outcomes are independent. Ex: Tossing a coin does not affect future tosses. A Bernoulli Trial DOES NOT have to have a 50-50 chance of success. Ex. Getting a "6" on a die roll is also binomial (you either get a 6 or you don't).

mutually inclusive events

Independent events occurring individually. Events are mutually inclusive if their outcomes may occur at the same time. Ex: The Venn diagram below illustrates the event of selecting a single card that is an ace or a diamond from a deck of cards. If events A and B are inclusive, the probability that either A or B occurs is the sum of their individual probabilities minus the probability of both occurring (so that this part isn't counted twice).

mutually exclusive events

Independent events occurring individually.Events are mutually exclusive if it is not possible for them to occur at the same time. One event or the other can occur, but not both. Ex: Drawing a 2 of diamonds or a 2 of spades on a single draw. to find probability of mutually exclusive events, you sum the individual probabilities P(A or B) = Pa + Pb

bayes theorem formula

P(A|D) = ( P(A) * P(D|A)) / (P(A)*P(D|A)+P(B)P(D|B)+P(C)P(D|C))

binomial process

The combination of many Bernoulli Trials using the Binomial Probability Formula

Bayes Theorem

The formula for the intersection of two dependent events, P(A ∩ B) = P(B)• P(A|B) can be rewritten as: P(A ∩ B) = P(A)• P(B|A) Then, by the transitive property: P(B)• P(A|B) = P(A)• P(B|A) Bayes Theorem is useful when you don't know the P(A ∩ B) term, but have enough other information that you can calculate it.

law of large numbers

The more trials that are conducted, the closer the results become to the theoretical probability

observed relative frequency

The number of times an event will occur divided by the number of opportunities. Similar to probability, used specifically when performing experiments. fx = nx/n b/w 0 and 1, fraction, percent, decimal total frequency of all possible events totals 1

conditional probability

The probability of an event given the knowledge that some preceding event has already occurred. Conditional probabilities occur when you are looking for the intersection of two or more dependent events: P(A and B) = P(B) • P(A|B) P(A and B) equals the probability of B times the probability of A given that B has occurred.

some things to remember

When asked to find a conditional probability, you will be given additional information about an event that has already occurred. This additional information helps to reduce the sample space that you would otherwise be working with and helps to improve your chance of success.

theoretical probability

a way of communicating the belief that an event will occur. Expressed as a number between 0 and 1. fraction, percent, decimal. Probability of an Event = # of ways event can succeed / total number of possible outcomes odds of an event probability of success / probability of failure: or # of successes/# of failures

experiment

an activity with observable results: test a brass sample to find its tensile strength

subjective

assumptions. i think that there is a 70% chance of rain today based on looking at the cloud coverage outside

theoretical

calculations based on mathematical theorems. no experimental component. using binomial probability formula to calculate the probability of flipping a coin 100 times and having it land heads up 57 of those times

independent events

events where the outcome of one DOES NOTaffect the outcome of the other. If events A and B are independent, the probability of both events occurring is: P(A and B) = PA∙PB P(A and B) = P(A ∩ B)

Dependent events

events where the outcome of one DOES affect the outcome of the other. If events A and B are dependent, the conditional probability of both events occurring is: P(A and B) = PA∙P(B|A) = PB∙P(A|B)

not the intersection (NAND) of a & b

everything but middle part

empirical

experimental observation - flipping a coin 100 times and counting the number of heads to determine the experimental probability of obtaining a heads

a minus b

only a

probability - NOT

probability of an independent event not occurring: 1 minus the probability of occurrence P=1-P(A)

binomial probability formula

px =( n! / (x! (n-x)!) ) (p^x)(q^n-x) or --> nCx p^x q^(n-x)

outcome/sample point

result of an experiment. when test is performed, the sample breaks at 850 pounds

sample space

set of all possible outcomes of an experiment. The brass sample might break under any load from 0 - 100 lbs or it may not break at all. The sample space are all of the load values at which it might break including the case where it may not break.

event

subset of a sample space. one possible event is that sample breaks at 200 pounds.