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Part 2
Chapter 5 Fundamentals of Probability
Random Experiments and Probability
An experiment is an activity or occurrence with an observable result.
Each repetition of an experiment is called a trial.
A possible result is called outcome or sample point.
The set of all possible outcomes for an experiment is the sample space.
A random experiment is an act which yields an outcome that belongs to a known set of
outcomes. The particular result that will be obtained is not known in advance of the act.
An event is any subset of a sample space.
An event with only one possible outcome is a simple event.
If an event E equals the sample space S, then E is a certain event.
If E= Ø, then E is an impossible event.
Tree Diagram is a graphical tool used to determine the sample space of random
experiments.
Venn Diagram is a way of graphically portraying the sample space and various events.
Set Operations for Events
Let A and B be events for a sample space S. then
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. We write A ∩ B for the intersection of A and B. A ∩ B consists
of all the sample points belonging to both A and B.
The Union of two events A and B is the event that occurs if either A or B or both occur 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.
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The complement of an event A is the event that occurs when A does not occur. That is, the
event consisting of all sample points that are not in event A. We denote the complement of A
by A’ or 𝐴𝑐 or 𝐴̅
Mutually Exclusive Events are events that do not share any sample point.
Events A and B are disjoint events if AnB = Ø
Probability
Let S be a sample space of equally likely outcomes, and let event A be a subset of S. The
probability of the event A, denoted by P (A), may be expressed as the ratio of the number of
elements in A to the total number of elements in the sample space S of the experiment.
P (A) =
𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠 𝑖𝑛 𝐴
𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠 𝑖𝑛 𝑆
=
𝑛(𝐴)
𝑛(𝑆)
Properties: 1) For any event A, 0 ≤ P (A) ≤ 1
2) If A = Ø, then P (Ø) = 0
3) If A = S, then P (A) = 1
4) P (A’) = 1 – P (A)
Addition Rule for Probability
For any events A and B from a sample space S, P (A U B) = P (A) + P (B) –P (A n B).
Probability of Union of two Mutually Exclusive Events.
If A and B are two mutually exclusive events, then: P (A U B) = P (A) + P(B)
(AnB = Ø )
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If k events are mutually exclusive, the probability that one of them will occur equals
the sum of their respective probabilities. In symbols,
P(A1 U A2 U A3 U⋯ U Ak) = P(A1) + P(A2) + P(A3) ⋯ P(Ak)
Complement rule
For any event A, P(A’) = 1- P(A) and P(A) = 1- P(A’) .
Odds
The odds of an event E, is a comparison of P (A) with P (A’).
1) The odds in favor of an event A is defined as the ratio of P(A) to P(A’),
P(A’) ≠ 0
𝑃(𝐴)
𝑃(𝐴′ )
=
𝑡ℎ𝑒 𝑝𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑡ℎ𝑒 𝑒𝑣𝑒𝑛𝑡 𝑤𝑖𝑙𝑙 𝑜𝑐𝑐𝑢𝑟
𝑡ℎ𝑒 𝑝𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑡ℎ𝑒 𝑒𝑣𝑒𝑛𝑡 𝑤𝑖𝑙𝑙 𝑛𝑜𝑡 𝑜𝑐𝑐𝑢𝑟
2) The odds against of an event A is defined as the ratio of P (A’) to
P (A),
𝑃(𝐴′)
the odds against A is the ratio: 𝑃(𝐴) , P(A) ≠ 0
Formula relating probabilities to Odds
If the odds are a to b that an event will occur, the probability of its occurrence is
P=
𝑎
𝑎+𝑏
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Conditional Probability
Conditional Probability is a probability calculated on a reduced sample space. This
reduced sample space is defined by a pre-established event (condition).
If P(B) ≠0, then the conditional probability of A relative to B, namely, the probability of A
given B, is
𝑃(𝐴∩𝐵)
P(𝐴|𝐵) = 𝑃(𝐵)
Multiplication Rule of Probability
𝑃(A ∩ B) = P(A) P(B|𝐴) or, equivalently, 𝑃(A ∩ B) = P(B) P(A|𝐵)
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.
Probability of Intersection of two Independent Events
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; that is,
𝑃(A ∩ B) = P(A) P(B)
The converse is also true: If 𝑃(A ∩ B) = P(A) P(B), then A and B are independent.
Contingency Table is a two-way table containing frequency data on two categorical variables.
Probability Tree is a tree diagram involving probabilities of given events.
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Chapter 4
The multiplication Principle, Permutations, and Combinations
Multiplication Principle
Suppose n choices must be made, with
m1 ways to make choice 1,
and for each of these,
m2 ways to make choice 2,
and so on, with
mn ways to make choice n.
Then there are
m1∙m2∙m3⋯mn
different ways to make the entire sequence of choices.
Factorial notation
If n is a natural number, the symbol n! (read “n-factorial” ) denotes the product of all
the natural numbers from n down to 1,
n! = n(n-1)(n-2) ⋯ (3)(2)(1).
By definition,
0! = 1
1! =1
A Permutation is an ordered arrangement in which r objects are chosen from n distinct
(different) objects and repetition is not allowed. The symbol nPr represents the number of
r objects selected from n objects
Number of Permutations of n distinct objects taken r at a time
The number of permutations of r objects from a set of n distinct objects is
nPr
𝒏!
= (𝒏−𝒓)!
The number of permutations of an n element set is n!
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A combination is a collection, without regard to order, of n distinct objects without
repetition. The symbol nCr represents the number of combinations of n distinct objects
taken r at a time
Number of Combinations of n objects taken r at a time
The number of ways in which r objects can be selected from a set of n distinct objects is
nCr
=
𝒏!
𝒓!(𝒏−𝒓)!
Permutations
Combinations
Different orderings arrangements of the r Each choice or subset of r objects gives 1
objects are different permutations.
combination. Order within the r object does
𝑛!
not matter.
nPr =
𝑛!
(𝑛−𝑟)!
nCr =
𝑟!(𝑛−𝑟)!
Clue words: arrangement, schedule, order
Order matters!
Clue words: group, committee, set, sample.
Order does not matter!
STA2023
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Part 2
CHAPTER 6 PROBABILITY DISTRIBUTIONS
Random Variable
A random variable is a function that assigns a real number to each outcome of an
experiment.
Types of random variables: Discrete and Continuous.
Discrete Random Variables are random variables defined on isolated real numbers. They
are typically used for counting.
Continuous Random Variables are random variables defined on a line interval of real
numbers. They are typically used for measuring.
Discrete Probability Distribution is a table, graph or formula assigning probabilities to
each value of a discrete random variable.
Probability Histogram is a graphical representation of a discrete probability distribution
associating the heights of rectangles with the given probabilities.
Probability Point Graph is a graphical representation of a discrete probability
distribution associating the heights of vertical lines with the given probabilities.
Expected Value
Suppose that the random variable X can take on n values x1, x2, x3, ⋯, xn.
Suppose also that the probabilities that these values occur are , respectively, p1, p2, p3, ⋯ pn.
Then the mean or expected value of a discrete random variable is
E(X) = x1 p1 + x2 p2 + x3 p3 + ⋯ + xn pn.
or
E(X) = ∑ 𝑥𝑃(𝑥)
Standard deviation of a discrete Random Variable
The standard deviation of a discrete random variable is 𝜎 = √∑ 𝑥 2 𝑃(𝑥) − 𝜇 2
The Binomial Distribution
Binomial experiment is a random experiment involving a number of identical and
independent trials in which there are only two possible outcomes (success and failure).
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Condition of a Binomial Experiment
1. The same experiment is repeated a fixed number of times.
2. There are only two possible outcomes: success and failure.
3. The probability of success for each trial is constant.
4. The repeated trials are independent.
Binomial random variable is a discrete random variable describing the number of
successes in a binomial experiment.
Binomial Probability
If p is the probability of success in a single trial of a binomial experiment, the
probability of x successes and n-x failures in n independent repeated trials of the
experiment is
f(x) = P(X= x) = nCxpx (1 – p)n-x for x = 0,1, 2,3, ⋯ , or n.
where
n = total number of trials
x =number of successes in n trials.
p = probability of success
p(X=x) denotes the probability of getting exactly x successes among the n trials
Parameters of the binomial probability distribution are the number of trials “n” and
the rate of success “p” (probability of success for each trial).
Mean (or Expected Value) and Standard deviation of a Binomial Distribution
A binomial experiment with n independent trials and probability of success p has a
mean and a standard deviation given by the formulas
E(X) = np and 𝜎 = √𝑛𝑝(1 − 𝑝)