Probability

Basic Rules for Computing Probability Rule 1: Relative Frequency Approximation of Probability

Conduct (or observe) a procedure, and count the number of times event A actually occurs. Based on these actual results, P(A) is approximated as follows: P(A) = # of times A occurred/ # of times procedure was repeated

Disjoint or Mutually Exclusive

Events A and B are DISJOINT (or MUTUALLY EXCLUSIVE) if they cannot occur at the same time. (That is, disjoint events do not overlap.)

Intuitive Addition Rule (COMPOUND EVENT)

To find P(A or B): -find the sum of the number of ways event A can occur -the number of ways event B can occur -add in such a way that every outcome is COUNTED ONLY ONCE -P(A or B) is equal to that sum, divided by the total number of outcomes in the sample space

Finding the Probability of "At Least One"

To find the probability of "at least one" of something, calculate the probability of "none", then subtract that result from 1. That is, ---> P(at least one) = 1 - P(none)

Complementary Events

P(A) and P(A`) are disjoint It is impossible for an event and its complement to occur at the same time.

Compound Event

*any event combining 2 or more simple events NOTATION: P(A or B) = P (in a single trial, event A occurs or event B occurs or they both occur)

*EVENT *SIMPLE EVENT *SAMPLE SPACE

-any collection of results or outcomes of a procedure -an outcome or an event that cannot be further broken down into simpler components -for a procedure consists of the set of all possible outcomes (that cannot be broken down any further) all possible "simple" events; outcomes

Addition Rule

-for the probability that either event A occurs or event B occurs (or they both occur) as the single outcome of a procedure. -for probabilities that can be expressed as P(A or B) -The key word in this section is "or" It is the inclusive or, which means either one or the other or both.

Conditional Probability

A ---- ---- of an event is a probability obtained with the additional information that some other event has already occurred. P(B|A) denotes the conditional probability of event B occurring, given that event A has already occurred, and it can be found by dividing the probability of events A and B both occurring by the probability of event A: P(B / A) = P(A and B) / P(A)

Law of Large Numbers

As a procedure is repeated again and again, the relative frequency probability of an event tends to approach the actual probability.

Rule 2: Classical Approach to Probability (Requires Equally Likely Outcomes)

Assume that a given procedure has n different simple events and that each of those simple events has an equal chance of occurring. If event A can occur in s of these n ways, then P(A) = s/n= number of ways A can occur/ number of different simple events

Conditional probability

Find the probability of an event when we have additional information that some other event has already occurred.

Probability of "at least one"

Find the probability that among several trials, we get at least one of some specified event. -"At least one" is equivalent to "one or more." -The COMPLIMENT of getting "at least one" item of a particular type is that you get "no" items of that type.

Notation for Probabilities

P - denotes a probability. A, B, and C - denote specific events. P(A) - denotes the probability of event A occurring.

Formal Addition Rule (COMPOUND EVENT)

P(A or B) = P(A) + P(B) - P(A and B) where P(A and B) denotes the probability that A and B BOTH OCCUR AT THE SAME TIME as an outcome

Rule of Complementary Events

P(A) + P(A`) = 1 P(A`) = 1 - P(A) P(A) = 1 - P(A`)

Intuitive Approach to Conditional Probability

The conditional probability of B given A: ~assume that event A has occurred ~then calculate the probability that event B will occur.

General Rule for a Compound Event

When finding the probability that event A occurs or event B occurs, find the total number of ways A can occur and the number of ways B can occur, but find that total in such a way that no outcome is counted more than once.