Statistics chapter 4

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

Complementary Events

The complement of event A, consists of all outcomes in which the event A does not occur.

Intuitive Approach to Conditional Probability

The conditional probability of B given A can be found by assuming that event A has occurred and then calculating the probability that event B will occur.

Compound Event Intuitive Addition Rule

To find P(A or B), find the sum of the number of ways event A can occur and the number of ways event B can occur, adding 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.

independent

Two events A and B are independent if the occurrence of one does not affect the probability of the occurrence of the other. (Several events are similarly independent if the occurrence of any does not affect the probabilities of the occurrence of the others.)

dependent

Two events are dependent if the occurrence of one of them affects the probability of the occurrence of the other, but this does not necessarily mean that one of the events is a cause of the other

Rounding Off Probabilities

When expressing the value of a probability, either give the exact fraction or decimal or round off final decimal results to three significant digits. (Suggestion: When a probability is not a simple fraction such as 2/3 or 5/9, express it as a decimal so that the number can be better understood.) All digits are significant except for the zeros that are included for proper placement of the decimal point.

Intuitive Multiplication Rule

When finding the probability that event A occurs in one trial and event B occurs in the next trial, multiply the probability of event A by the probability of event B, but be sure that the probability of event B takes into account the previous occurrence of event A.

Compound Event General Rule

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.

Simulations

a process that behaves in the same ways as the procedure itself so that similar results are produced.

simulation of a procedure

a process that behaves the same way as the procedure, so that similar results are produced.

Basic Rules for Computing Probability

1. Relative Frequency Approximation of Probability 2. Classical Approach to Probability (Requires Equally Likely Outcomes) 3. Subjective Probabilities

Odds types

1. actual odds against 2. actual odds in favor 3. payoff odds

Tree Diagrams

A tree diagram is a picture of the possible outcomes of a procedure, shown as line segments emanating from one starting point. These diagrams are sometimes helpful in determining the number of possible outcomes in a sample space, if the number of possibilities is not too large.

Probability Limits

Always express a probability as a fraction or decimal number between 0 and 1. The probability of an impossible event is 0. The probability of an event that is certain to occur is 1. For any event A, the probability of A is between 0 and 1 inclusive.

Definition

An event is unlikely if its probability is very small, such as 0.05 or less. An event has an usually low number of outcomes of a particular type or an unusually high number of those outcomes if that number is far from what we typically expect.

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

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

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.)

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.

Multiplication Rule for Several Events

In general, the probability of any sequence of independent events is simply the product of their corresponding probabilities.

actual odds in favor

The actual odds in favor of event A occurring are the ratio, which is the reciprocal of the actual odds against the event. If the odds against A are a:b, then the odds in favor of A are b:a.

Compound Event Formal Addition Rule

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 in a trial of a procedure.

Subjective Probabilities

P(A), the probability of event A, is estimated by using knowledge of the relevant circumstances.

P(A and B)

P(event A occurs in a first trial and event B occurs in a second trial)

Treating Dependent Events as Independent

Some calculations are cumbersome, but they can be made manageable by using the common practice of treating events as independent when small samples are drawn from large populations. In such cases, it is rare to select the same item twice.

actual odds against

The actual odds against event A occurring are the ratio, usually expressed in the form of a:b (or "a to b"), where a and b are integers having no common factors.

payoff odds

The payoff odds against event A occurring are the ratio of the net profit (if you win) to the amount bet.

Simple Event

an outcome or an event that cannot be further broken down into simpler components

Event

any collection of results or outcomes of a procedure

Compound Event

any event combining 2 or more simple events

A, B, and C

denote specific events.

P

denotes a probability.

P(A)

denotes the probability of event A occurring.

Sample Space

for a procedure consists of all possible simple events; that is, the sample space consists of all outcomes that cannot be broken down any further

"At least one"

is equivalent to "one or more."

complement of getting at least one item of a particular type

is that you get no items of that type.

P(B | A)

the probability of event B occurring after event A has already occurred.