3.1 Terminology
- Page ID
- 36475
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Learning Objective:
In this section, you will:
• Understand the fundamentals of probability
Probability is a number between zero and one, inclusive, that gives the likelihood that a specific event will occur.
• The probability of any outcome is the long-term relative frequency of that outcome.
• P(A) = 0 means the event A can never happen.
• P(B) = 1 means the event B is certain to happen.
• P(C) = 0.5 means that event C is equally likely to happen or not happen.
An experiment is a planned activity carried out under controlled conditions.
An outcome is particular result of an experiment.
The sample space is the set of all possible outcomes of an experiment.
An event is a collection of results or outcomes of an experiment.
Equally likely means that each outcome of an experiment has the same probability.
To calculate the probability of an event A when all outcomes in the sample space are equally likely, count the number of outcomes for event A and divide by the total number of outcomes in the sample space.
Notation:
• P represents a probability
• A, B, and C represent specific events
• P(A) represents the probability of event A occurring
• P(A) = (number of outcomes for event A)/ (total number of outcomes in the sample space)
The law of large numbers states that as the number of repetitions of an experiment is increased, the relative frequency obtained in the experiment tends to become closer and closer to the theoretical probability.
Example 1
Roll a standard six-sided die once.
Sample space: S =
Identify each of the following events with a subset of S and compute its probability.
a. Event A = Roll an even number
b. Event B = Roll a number larger than 4
c. Event C = Roll a 2
d. Event D = Roll a 9
e. Event E = Roll number less than 10
The complement of event A consists of all outcomes that are NOT in A. Which is denoted as A′ (read "A prime") or A̅ .
P(A’)= P(B’)= P(C’) =
Notice that P(A) + P(A′) = 1.
"OR" Event
An outcome is in the event A OR B if the outcome is in A or is in B or is in both A and B
Example 2: let A = {1, 2, 3, 4, 5} and B = {4, 5, 6, 7, 8}.
A OR B =
"AND" Event
An outcome is in the event A AND B if the outcome is in both A and B at the same time.
Example 3: let A and B be {1, 2, 3, 4, 5} and {4, 5, 6, 7, 8}.
A AND B =
“GIVEN” Event
The conditional probability of A given B is written P(A|B). P(A|B) is the probability that event A will occur given that the event B has already occurred. A conditional reduces the sample space. We calculate the probability of A from the reduced sample space B.
Example 4: Suppose we toss one fair, six-sided die. The sample space S = {1, 2, 3, 4, 5, 6}. Let A = face is 2 or 3 and B = face is even (2, 4, 6).
A GIVEN B =
Mutually Exclusive Events
A and B are mutually exclusive events if they cannot occur at the
same time. This means that A and B do not share any outcomes.
Example 5: Suppose the sample space S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. Let A = {1, 2, 3, 4, 5}, B =
{4, 5, 6, 7, 8}, and C = {7, 9}.
A AND B =
A AND C =
For more information and examples see online textbook OpenStax Introductory Statistics pages 176- 180.
“Introduction to Statistics” by OpenStax, used is licensed under a Creative Commons Attribution License 4.0 license