Chapter 4

Intuitive Approach to Conditional Probability

Finding P(B|A): The conditional probability of B occurring given that A has occurred can be found by assuming that event A has occurred and the calculating the probability that event B will occur.

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.

With Replacement or Without Replacement

Sampling with replacement: selections are independent events. Sampling without replacement: selections are dependent events.

Notes for 3 Different Odds

The actual odds against and the actual odds in favor are calculated with the actual likelihood of some event, but the payoff odds describe the relationship between the bet and the amount of the payoff. The actual odds correspond to actual probabilities of outcomes, but the payoff odds are set by racetrack and casino operators. Racetracks and casinos are in business to make a profit, so the payoff odds will not be the same as the actual odds.

4 Steps to Find the Probability of Getting At Least One of Some Event

1. Let A = getting at least one of some event; 2. Then A-bar = getting none of the event being considered; 3. Find P(A-bar) = probability that event A does not occur; 4. Subtract the result from 1, that is, evaluate this, P(at least one occurrence of event A) = 1 - P(no occurrences of event A).

3 Different Approaches to Finding the Probability

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

3 Different Odds

1. The actual odds against event A occurring are the ratio P(A-bar)/P(A), usually expressed in the form of a:b (or "a to b"), where a and b are integers having no common factors. 2. The actual odds in favor of event A occurring are the ratio P(A)/P(A-bar), which is the reciprocal of the actual odds against that event. If the odds againA are a:b, then the odds in favor of A are b:a. 3. The payoff odds against event A occurring are the ratio of net profit (if you win) to the amount bet: payoff odds against event A = (net profit):(amount bet).

5 Important Principles and Notation for Probability

1. The probability of an event is a fraction or decimal number between 0 and 1 inclusive. 2. The probability of an impossible event is 0. 3. The probability of an event that is certain to occur is 1. 4. Notation: The probability of event A is denoted by P(A). 5. Notation: The probability of event A does not occur is denoted by P(A-bar).

Exception: Treating Dependent Events as Independent

5% Guideline for Cumbersome Calculations: When calculations with sampling are very cumbersome (7 or more selections) and the sample size is no more than 5% of the size of the population, treat the selections as being independent even if they are actually dependent.

Compound Event

A compound event is any event combining two or more simple events.

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. P(B|A) 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)

Event

A particular outcome of the sample space.

Simple Event

A simple event is an outcome or an event that cannot be further broken down into simpler components.

Unlikely Events and Unusual Events

An event is unlikely if its probability is very small, such as 0.05 or less. An event has an unusually 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. Unlikely: Small probability (such as 0.05 or less) Unusual: Extreme result (number of outcomes of a particular type is far below or far above the typical values, by associating "unusual" with extreme outcomes, we are consistent with the range rule of thumb and the use of z scores for identifying unusual values.)

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. With increased repetition of the experiment (procedure), the relative frequency probability approaches classical probability.

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 occur / number of different simple events = s / n When using the classical approach, always verify that the outcomes are equally likely. Example: Winning the grand prize in a lottery by selecting 6 numbers.

Combinations Committee (Order doesn't matter)

Combinations of items are arrangements in which different sequences of the same items are not counted separately. For example, with the letters {a, b, c}, the arrangement of abc, acb, bac, bca, cab, and cba are all considered to be same combination.

Relative Frequency Approximation of Probability

Conduct or observe a procedure, and count the number of times that event A actually occurs (controled). Based on these actual results, P(A) is approximated as: P(A) = number of times A occurred / number of times the procedure was repeated Example: An individual car crashes in a year.

Disjoint

Events A and B are disjoint or mutually exclusive if they cannot occur at the same time. Disjoint events do not overlap. The addition rule is simplified when the events are disjoint.

Notation for Probabilities

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

Formal Multiplication Rule

P(A and B) = P(A) • P(B|A)

Multiplication Rule Notation

P(A and B) = P(event A occurs in a first trial and event B occurs in a second trial), P(B|A) represents the probability of event B occurring after it is assumed that event A has already occurred.

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.

Notation for Addition Rule

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

Subjective Probabilities

P(A), the probability of event A, is estimated by using knowledge of the relevant circumstances. Note that the classical approach requires equally likely outcomes. If the outcomes are not equally likely, we must use the relative frequency approximation or we must rely on our knowledge of the circumstances to make an educated guess. Example: A passenger dying in a plane crash. Probable: A probability on the order of 0.00001 or greater. Improbable: A probability on the order of 0.00001 or less.

Permutations Position (Order matters)

Permutations of items are arrangements in which different sequences of the same items are counted separately. For example, with the letters {a, b, c}, the arrangements of abc, acb, bac, bca, cab, and cba are all counted separately as six different permutations.

Complementary Event

The complement of event A, denoted by "A-bar", consists of all outcomes in which event A does not occur.

Factorial Symbol

The factorial symbol (!) denotes the product of decreasing positive whole numbers. For example, 4! = 4 • 3 • 2 • 1 = 24. By special definition, 0! = 1.

Sample Space

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

Intuitive Addition Rule

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

Intuitive Multiplication Rule

To find the probability that event A occurs in one trial and event B occurs in another 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.

Independent or Dependent Events

Two events A and B are independent if the occurrence of one does not affect the probability of the occurrence of the other. If A and B are not independent, they are said to be dependent.

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. All digits in a number are significant except for the zeros that are included for proper placement of the decimal point.

Fundamental Counting Rule

m • n = Number of ways that two events can occur, given that the first event can occur m ways and the second event can occur n ways.

Permutations Rule (When Some Item Are Identical to Others)

n! / n1!n2!...nk! = Number of different permutations (order counts) when n items are available and all n are selected without replacement, but some of the items are identical to others: n1 are alike, n2 are alike, ..., and nk are alike.

Factorial Rule

n! = Number of different permutations (order counts) of n different items when all n of them are selected.

Combinations Rule

nCr = n! / (n-r)!r! = Number of different combinations (order does not count) when n different items are available, but only r of them are selected without replacement.

Permutations Rule (When All of the Items Are Different)

nPr = n! / (n-r)! = Number of different permutations (order counts) when n different items are available, but only r of them are selected without replacement.