Ch 4: Probability

Fundamental Counting Rule

For a sequence of two events in which the first event can occur m ways and the second event can occur n ways, the events together can occur a total of m n ways.

Rare Event Rule for Inferential Statistics:

If, under a given assumption, the probability of a particular observed event is extremely small, we conclude that the assumption is probably not correct.

actual odds against

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

Complements: The Probability of "At Least One"

"At least one" is equivalent to "one or more." The complement of getting at least one item of a particular type is that you get no items of that type.

payoff odds against event A

(net profit) : (amount bet)

Conditional Probability

A conditional probability 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) over P(A)

Probability Limits

Always express a probability as a fraction or decimal number between 0 and 1.

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)= number of ways A can occur over number of different simple events

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: # of times A occurred over # 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.)

Complementary Events

It is impossible for an event and its complement to occur at the same time.

Formal Multiplication Rule

P(A and B) = P(A) • P(B|A)

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.

Rule 3: Subjective Probabilities

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

Complementary Events

The complement of event A, denoted by A (bar) , consists of all outcomes in which the event A does not occur.

Conditional Probability Important Principle

The probability for the second event B should take into account the fact that the first event A has already occurred.

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.

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).

Confusion of the Inverse

To incorrectly believe that P(A|B) and P(B|A) are the same, or to incorrectly use one value for the other, is often called confusion of the inverse.

Dependent Events

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.

Conditional Probability Key Point

We must adjust the probability of the second event to reflect the outcome of the first event.

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.)

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.

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.

tree diagram

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.

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. Notation: P(A or B) = P (in a single trial, event A occurs or event B occurs or they both occur)

A, B, and C

denote specific events.

P

denotes a probability.

P(A)

denotes the probability of event A occurring.

!

denotes the product of decreasing positive whole numbers. 0!=1

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

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.)

The actual odds in favor

of event A occurring are the ratio P(A[bar])/P(A), 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.