Chapter 3

Sample Point

A sample point is the most basic outcome of an experiment.

Mutually Exclusive

Events A and B are mutually exclusive if A ∩ B contains no sample points - that is, if A and B have no sample points in common. For mutually exclusive events, P (A ∩ B) = 0

Multiplicative Rule of Probability

P (A ∩ B) = P (A) P (B | A) or, equivalently, P (A ∩ B) = P (B) P (A | B)

Sample Space

The Sample Space of an experiment is the collection of all its sample points.

Rule of complements

The sum of the probabilities of complementary events equals 1; that is, P (A) + P (A^C) = 1.

Additive Rule of Probability

The probability of the union of events A and B is the sum of the probability of event A and the probability of event B, minus the probability of the intersection of events A and B; that is: P (A ∪ B) = P (A) + P (B) - P (A ∪ B)

Conditional Probability Formula

To find the conditional probability that event A occcurs given event B occurs, divide the probability that both A and B occur by the probability that B occurs; that is,

Compound Events

An event can often be viewed as a composition of two or more other events. Such events which are called compound events, can be formed (composed) in two ways. (Union) (Intersection)

Event

An event is a specific collection of sample points.

Experiment

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

Probability of Intersection of Two Independent Events

If events A and B are independent, then the probability of the intersection of A and B equals the product of the probabilities of A and B; that is, P (A ∩ B) = P (A) P (B) This converse is also true: P (A ∩ B) = P (A) P (B), then events A and B are independent.

Complement

The complement of an event A is the event that A does not occur - that is, the event consists of all sample points that are not in event A. We denote the complement of A by A^c.

Steps for Calculation Probabilities of Events

1. Define the experiment; this 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 probability of the event.

Independent Events

Events A and B are independent events if the occurrence of B does not alter the probability that A has occurred; that is, events A and B are independent if P (A | B) = P (A) When events A and B are independent, it is also true that P (B | A) = P (B) Events that are not independent are said to be dependent.

Permutations Rule

Given a single set of N different elements, you wish to select n elements from the N and arrange them within n positions. The number of sifferent permutations of the N elements taken n at a time is denoted by P (N n) and is equal to (See Graph) where n! = n ( n - 1) (n - 2)... and is called n factorial.

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 probability of A and the probability of B; that is, P (A ∪ B) = P (A) + P (B).

Probability Rules for Sample Points

Let p i represent the probability of ample point i. Then 1. All sample point probabilities must lie between 0 and 1 (i.e. 0 ≤ p i ≤ 1) 2. The probabilities of all the sample points within a sample space must sum to 1 (i.e. ∑ p i = 1)

Combination Rule

Suppose a sample of n elements is to be drawn without replacement from a set of N elements. Then the number of different samples possible is denoted by ( N n) and is equal to: substitute: n for N and k for n

Partitions Rule

Suppose you wish to partition a single set of N different elements in k sets, with the first set containing n ₁ elements, the second n₂ elements, ... , and the kth sent containing n k elements. Then the number of different partitions is: N! / (n₁! n₂! ... nk!), where n₁ + n ₂ + ... n k = N

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.

Intersenction

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

Union

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

The Multiplicative Rule

You have k sets of elements, n ₁ in the first set, n₂ in the second set, ... , and n k in the kth set. Suppose you wish to form a sample of k elements by taking one element from each of the k sets. Then the number of different samples that can be formed is the product n₁n₂n₃....nk