CH. 3- Probability

Steps for Calculating Probabilities of Events

1. Define the experiment; that is, describe the process used to make an observation and the type of observation that will be recorded. 2. List the Sample points. 3. Assign probabilities to the sample points. 4. Determine the collection of sample points contained in the event of interest. 5. Sum the sample point probabilities to get the event probability

Conditional Probability

A probability that reflects additional knowledge that might affect the likelihood of the outcome of an experiment, so we need to alter the probability of an event of interest. [like if told die already cast, want to know P of observing an even #, which is 1/2, except that told it is 3 or less

Compound Event

Contains two or more sample points (events)

Unconditional Probability

Event probabilities which gives the relative frequencies of the occurrences of the events when the experiment is repeated a very large number of times. Called unconditional because no special conditions are assumed other than those that define the experiment.

Probability of Union of Two Mutually Exclusive Events

If two events A and B are mutually exclusive, the probability of the union of A and B equals the sum of the probabilities of A and B; that is, P(A U B) = P(A) + P(B)

Probability Rules for Sample Points

Let pi represent the probability of sample point i. Rule 1: All sample point probabilities MUST lie between 0 & 1 (i.e., 0 <= pi <=1). Rule 2: The probabilities of all the sample points within a sample space MUST sum to 1 (i.e., (sum not.) Epi=1).

Probability vs. Statistics

Probability is the reverse of Statistics. In Statistics (one branch), one uses the SAMPLE information to infer the probable nature of the population. In Probability, one uses the POPULATION information to infer the probable nature of the sample

Venn Diagram

Sample space showed as a closed figure (box) (labeled) containing all possible sample points. Each sample point represented bu solid dot (a point) and labeled accordingly.

Union Of Two Events

The Union of two events A and B is the event that occurs if either A or B or both occur on a single performance of the experiment. We denote the union of events A and B by the symbol A U B. A U B consists of all the sample points that belong to A or B or Both.

Intersection of Two Events

The intersection of two events A and B is the event that occurs if both A and B occur on single performance of the experiment. We write A ∩ B for the intersection of A and B. A ∩ B consists of all the sample points belonging to both A and B.

Probability of an Event

The probability of an event A is calculated by summing the probabilities of the sample points in the sample space for A.

Additive Rule of Probability

The probability of the union of events A and B is the sum of the probabilities of events A and B minus the probability of the intersection of events A and B: P(A U B)= P(A)+P(B)- P(A ∩ B). [P(A ∩ B)= P(A) x P(B)

Rule of Compliments

The sum of the probabilities of complementary events equals 1: P(A)+P(A^c)=1. Often easier to calculate than the event itself. THEN can calculate P(A) by reordering: P(A)=1-P(A^c).

Intersection Symbol

U

Union Symbol

U

Probability

Use the population information to infer the probable nature of the sample

Mutually Exclusive

[A and B don't intersect on the Venn Diagram[]. Events A and B are mutually exclusive if A ∩ B contains no sample points- that is, A and B have no sample points together.

Complementary events

[A bar, meaning everything except]. The Complement of an event A is the event that A does NOT occur- that is , the event consisting of all sample points that are not in event A. We denote the complement of A by A^c (me- A Bar)

Event

a specific collection of sample points.

Experiment

an act or process of observation that leads to a single outcome that cannot be predicted with certainty

Sample Space

collection of all the sample points in a space

Simple Event

contains only a single sample point

Tree Diagram

for listing the sample points

Factorial Symbol

n!= n x (n-1) x (n-2) ... 3 x 2 x 1; Example: 5!= 5x4x3x2x1

Combinations Rule

page 126 fill in later

Sample Point

the most basic outcome of an experiment. observing the outcome of an experiment is similar to selecting a sample from a population