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Chapter 8
Probability: The Mathematics of
Chance
Definitions
• Randomness - A phenomenon is said to be
random if individual outcomes are
uncertain but long-term patterns of many
individual outcomes is predictable.
• Probability – proportion of times the
outcome would occur in a very long series
of repetitions.
More Definitions
• Sample Space – the set of all possible outcomes.
• Event – any outcome or any set of outcomes of a
random phenomenon. That is, an event is a subset
of the sample space.
• Probability model – mathematical description of a
random phenomenon consisting of two parts: as
sample space S and a way of assigning
probabilities to events
Probability Rules
• Rule 1 – The probability P(A) of any event A
satisfies 0 ≤ P(A) ≤ 1.
• Rule 2 – If S is the sample space in a probability
model, then P(S) = 1.
• Rule 3 – Two events A and B are disjoint if they
have no outcomes in common and so can never
occur together. If A and B are disjoint,
P(A or B) = P(A) + P(B)
• Rule 4 – The Complement of any event A is the
event that A does not occur, written as Ac.
P(Ac) = 1 – P(A)
Discrete Probability
Model
• A probability model with a finite sample
space is called discrete. To assign
probabilities in a discrete model, list the
probability of all the individual outcomes.
These probabilities must be numbers
between 0 and 1 and must have a sum of 1.
• The probabilities of any event is the sum
of the probabilities of the outcomes
making up the event.
Equally Likely Outcomes
• If a random phenomenon has k
possible outcomes, all equally likely,
then each individual outcome has
probability 1/k. The probability of
any event A is
count of outcomes in A count of outcomes in A
P( A) 

count of outcomes in S
k
Counting Arrangements
of Distinct Items
• Rule 1 – Suppose we have a collection
of n distinct items. We want to
arrange k of these items in order,
and the same item can appear several
times in the arrangement. The
number of possible arrangements is
n n
 n  nk
• Rule 2 – Suppose we have a collection of n
distinct items. We want arrange k of
these items in order, and any item can
appear no more than once in the
arrangement. The number of possible
arrangements is
n  (n  1) 
 (n  k  1)
Continuous Probability
Models
• A density curve is a curve that
– Is always on or above the horizontal axis
– Has area exactly 1 underneath it.
• A continuous probability model assigns
probabilities as areas under a density
curve. The area under the curve and above
any range of values is the probability of an
outcome in that range.
The mean and standard
deviation of a probability
model
• The mean of a discrete probability model
– Suppose that the possible outcomes x1 , x2 , , xk
in a sample space S are numbers and that p j
is the probability of outcome x j . The mean µ of
the probability model is
  x1 p1  x2 p2 
 xk pk
Law of Large Numbers
• Observe any random phenomenon having numerical
outcomes with finite mean µ. According to the
Law of Large Numbers, as the random phenomenon
is repeated a large number of times,
– The proportion of trials on which each outcome occurs
gets closer and closer to the probability of that outcome,
and
– The mean x of the observed values gets closer and closer
to µ.
Standard Deviation of a
Discrete Probability Model
• Suppose that the possible outcomes x1 , x2 ,
in a sample space S are numbers and that
is the probability of outcome x . The
variance σ2 of the probability model is
, xk
pj
j
 2  ( x1   ) 2 p1  ( x2   ) 2 p2 
 ( xk   ) 2 pk
The standard deviation σ is the square root
of the variance.
Central Limit Theorem
• Draw a SRS of size n from any large
population with mean µ and finite standard
deviation σ. Then
– The mean of the sampling distribution of x is µ.
– The standard deviation of the sampling
distribution of x is  n .
– The central limit theorem says that the
sampling distribution of x is approximately
normal when the sample size n is large.