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Probability Concepts
Sunday, March 6, 2016

12:56 PM

Events, Sample Spaces, and Probability

• We use the population information to infer the probable nature of the sample
• An experiment is an act of process of observation that leads to a single outcome
that cannot be predicted with certainty
• A sample point is the most basic outcome of an experiment
• The sample space of an experiment is the collection of all its sample points
• The probability of a sample point is a number between 0 and 1 inclusive that
measures the likelihood that the outcome will occur when the experiment is
performed
Probability Rules for Sample Points
Let pi represent the probability of sample point i
All sample point probabilities must lie between 0 and 1
The probabilities of all sample points within a sample space must sum to 1
• An event is a specific collection of sample points
○ A simple event contains only a single sample point
○ A compound event contains two or more sample points
• The probability of an event A is calculated by summing the probabilities of the
sample points in the sample space for A
• Combinations Rule: a sample of n elements is to be drawn from a set of N
elements
○ The number of different samples possible is denoted by (N over n)
○ (N over n) = N!/n!(N - n)!
○ Applies to situations in which the experiment calls for selecting n
elements from a total of N elements without replacing each element
before the next is selected

Unions and Intersections

• The union of two events A and B is the event that occurs if either A or B or both
occur in a single performance of the experiment
○ Denoted A⌣B
• The intersection of two events A and B is the event that occurs if both A and B
occur on a single performance of the experiment
○ Denoted A⌢B

occur in a single performance of the experiment
○ Denoted A⌣B
• The intersection of two events A and B is the event that occurs if both A and B
occur on a single performance of the experiment
○ Denoted A⌢B

Complementary Events

• The complement of an event A is the event that A does not occur
○ The event consisting of all sample points that are not in A
○ Denoted A by A^c
• The sum of the probabilities of complementary events equals 1
○ P(A) + P(A^c) = 1

The Additive Rule and Mutually Exclusive Events

• The probability of the union of events A and B is the sum of the probabilities of
events A and B minus the probability of the intersection of events A and B
○ P(A⌣B) = P(A) + P(B) - P(A⌢B)
• Events A and B are mutually exclusive is A⌢B contains no sample points
○ A and B have no sample points in common
• If two events A and B are mutually exclusive, the probability of the union of A
and B equals the sum of the probabilities of A and B
○ P(A⌣B) = P(A) + P(B)

Conditional Probability

• Unconditional Probabilities: no special conditions are specified
• A probability that reflects additional knowledge that may affect the outcome of
an experiment is called the conditional probability of the event
• To find the conditional probability that event A occurs given that event B occurs,
divide the probability that both A and B occur by the probability that B occurs
○ P(A/B) = P(A⌢B)/P(B)

The Multiplicative Rule and Independent Events

• A probability that reflects additional knowledge that may affect the outcome of
an experiment is called the conditional probability of the event
• To find the conditional probability that event A occurs given that event B occurs,
divide the probability that both A and B occur by the probability that B occurs
○ P(A/B) = P(A⌢B)/P(B)

The Multiplicative Rule and Independent Events

• P(A⌢B) = P(A)*P(B/A) or equivalently, P(A⌢B) = P(B)*P(A/B)
• Events A and B are independent events if the occurrence of B does not alter the
probability that A has occurred
○ Independent if P(A/B) = P(A) or
○ P(B/A) = P(B)
○ Events that are not independent are said to be dependent
• Cannot be shown by a Venn Diagram
• Mutually exclusive events are dependent events because P(A) doesn't equal
P(A/B)
• If events A and B are independent, the probability of the intersection of A and B
equals the product of the probabilities of A and B
○ P(A⌢B) = P(A)*P(B)

Bayes' Rule

• Given k mutually exclusive and exhaustive events, B1, B2,
...
P(Bk) =1, and an observed event A, then
○ P(Bi/A) = P(Bi⌢A)/P(A) = P(Bi)*P(A/Bi) / P(B1)*P(A/B1) + P(B2)*P(A/B2)
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