Chapter 4
Probability experiment:
- chance process that leads to well-defined results called outdoes
Outcome:
- result of a single trial of a probability experiment
Sample space:
- set of all possible outcomes of a probability experiment
Event:
- consists of a set of outcomes of a probability experiment
Single event:
- event with one outcome
Compound event:
- event of getting an odd number
Equally likely events:
- events that have the same probability of occurring
Venn diagrams:
- used to present probabilities pictorially
Rounding rule for probabilities:
- should be expressed as reduced fractions or 2-3 decimal places.
Three basic interpretations of probability:
- Classical:
uses sample spaces to determine numerical probability that an event will happen
assumes that all outcomes in the sample space are equally likely to occur
Forumla: Four basic probability rules
P(E) probability of an event E 1) P(E) is a number between & including 0 & 1
n(E) number of outcomes in E 2) If an event E can’t occur, then P(E) = 0
n(S) total number of outcomes in sample space S 3) If an event E is certain, then P(E) = 1
Then P(E) = n(E) / n(S) 4) The sum of the probabilities of the outcomes
in the sample space is 1
Complimentary event:
- the compliment of an event E is the set of outcomes in the sample space that are not included in
the outcomes of event E.
It is denoted by or E’
Rules for complimentary events:
P(E’) = 1 - P(E) P(E)= 1-P(E’) P(E’) + P(E) = 1
- Empirical/Relative frequency
relies on actual experience to determine likelihood of outcomes. uses the concept of frequency
distributions
Formula: P(E) = frequency of the class / total frequencies in the distributions
- Subjective
based on educated guesses or estimates, employing opinions & inexact information
, Additional rules for probability
Mutually exclusive events:
events that cannot occur at the same time
Rule 1:
P(A or B) = P(A) + P(B) , where A & B are mutually exclusive
Rule 2:
P (A or B) = P(A) + P(B) - P(A and B) , where A & B are not mutually exclusive
Multiplication rule & conditional probability
- used to find the probability of two or more events that occur in sequence
- event A & B are independent events if A occurs without affecting the probability of B occurring
If A & B are independent the, P(A and B) = P(A) * P(B)
Dependent events:
when outcome of the first event affects the outcomes of the second event to the point where the
probability changes
Conditional probability:
when event B in relationship to event A is the probability that event B occurs after event A has
already occurred P(B|A)
Formula:
P(B|A) = P(A and B) / P(A)
Fundamental counting rule:
used to determine the total number of outmodes in a sequence of events
Factorial forumla:
n! = n(n-1)(n-2)….1
0! = 1
Permutations:
is an arrangement of n objects in a specific order
Rule:
- arrangement of n objects in a specific order using r objects is written as nPr
Formula:
nPr = n! / (n-r)!
Combinations:
is a selection of distinct objects without regard to order
number of combinations of r objects selected from n objects is written as nCr
Formula:
nCr = n! / (n-r)!r!
Summary:
- Probability can be defined as the chance of an event occurring.
- The three types of probability are classical, empirical, and subjective.
- Classical probability uses sample spaces.
- Empirical probability uses frequency distributions and is based on observations.
- In subjective probability, the researcher makes an educated guess
- An event consists of one or more outcomes of a probability experiment
- Two events are said to be mutually exclusive if they cannot occur at the same time.
- Events can also be classified as independent or dependent.
- If events are independent, whether or not the first event occurs does not affect the probability of
the next event occurring.
- If the probability of the second event occurring is changed by the occurrence of the first event,
then the events are dependent.
.
Probability experiment:
- chance process that leads to well-defined results called outdoes
Outcome:
- result of a single trial of a probability experiment
Sample space:
- set of all possible outcomes of a probability experiment
Event:
- consists of a set of outcomes of a probability experiment
Single event:
- event with one outcome
Compound event:
- event of getting an odd number
Equally likely events:
- events that have the same probability of occurring
Venn diagrams:
- used to present probabilities pictorially
Rounding rule for probabilities:
- should be expressed as reduced fractions or 2-3 decimal places.
Three basic interpretations of probability:
- Classical:
uses sample spaces to determine numerical probability that an event will happen
assumes that all outcomes in the sample space are equally likely to occur
Forumla: Four basic probability rules
P(E) probability of an event E 1) P(E) is a number between & including 0 & 1
n(E) number of outcomes in E 2) If an event E can’t occur, then P(E) = 0
n(S) total number of outcomes in sample space S 3) If an event E is certain, then P(E) = 1
Then P(E) = n(E) / n(S) 4) The sum of the probabilities of the outcomes
in the sample space is 1
Complimentary event:
- the compliment of an event E is the set of outcomes in the sample space that are not included in
the outcomes of event E.
It is denoted by or E’
Rules for complimentary events:
P(E’) = 1 - P(E) P(E)= 1-P(E’) P(E’) + P(E) = 1
- Empirical/Relative frequency
relies on actual experience to determine likelihood of outcomes. uses the concept of frequency
distributions
Formula: P(E) = frequency of the class / total frequencies in the distributions
- Subjective
based on educated guesses or estimates, employing opinions & inexact information
, Additional rules for probability
Mutually exclusive events:
events that cannot occur at the same time
Rule 1:
P(A or B) = P(A) + P(B) , where A & B are mutually exclusive
Rule 2:
P (A or B) = P(A) + P(B) - P(A and B) , where A & B are not mutually exclusive
Multiplication rule & conditional probability
- used to find the probability of two or more events that occur in sequence
- event A & B are independent events if A occurs without affecting the probability of B occurring
If A & B are independent the, P(A and B) = P(A) * P(B)
Dependent events:
when outcome of the first event affects the outcomes of the second event to the point where the
probability changes
Conditional probability:
when event B in relationship to event A is the probability that event B occurs after event A has
already occurred P(B|A)
Formula:
P(B|A) = P(A and B) / P(A)
Fundamental counting rule:
used to determine the total number of outmodes in a sequence of events
Factorial forumla:
n! = n(n-1)(n-2)….1
0! = 1
Permutations:
is an arrangement of n objects in a specific order
Rule:
- arrangement of n objects in a specific order using r objects is written as nPr
Formula:
nPr = n! / (n-r)!
Combinations:
is a selection of distinct objects without regard to order
number of combinations of r objects selected from n objects is written as nCr
Formula:
nCr = n! / (n-r)!r!
Summary:
- Probability can be defined as the chance of an event occurring.
- The three types of probability are classical, empirical, and subjective.
- Classical probability uses sample spaces.
- Empirical probability uses frequency distributions and is based on observations.
- In subjective probability, the researcher makes an educated guess
- An event consists of one or more outcomes of a probability experiment
- Two events are said to be mutually exclusive if they cannot occur at the same time.
- Events can also be classified as independent or dependent.
- If events are independent, whether or not the first event occurs does not affect the probability of
the next event occurring.
- If the probability of the second event occurring is changed by the occurrence of the first event,
then the events are dependent.
.
