PRED 354 TEACH. PROBILITY &
STATIS. FOR PRIMARY MATH
Lesson 6
Conditional Probability
Question
Suppose that a certain precinct contains 350 voters, of
which 250 voters are Democrats and 100 are
Republicans. If 30 voters are chosen at random
from the precinct, what is the probability that
exactly 18 Democrats will be selected?
Question
If A, B, and D are three events such that
Pr( A B D)  0,7
what is the value of
Pr( Ac
Bc
Dc )
Question
If the letters s, s, s, t, t, t, i, i, a, c, are arranged in a
random order,
what is the probability that they will spell the word
“statistics”?
Conditional Probability
The updated probability of event A after we
learn that event B has occurred is the
conditional probability of A given B.
Pr( A B)
Pr( A B) 
Pr( B)
Conditional probability of A given that B has
occurred.
Example
Find the probability of a 1, given the occurrence of an
odd number, in the toss of a single die.
Independence
Two events, A and B, are said to be
independent , if
Pr( A B)  Pr( A).Pr( B)
Otherwise, the
dependent.
events
are
said
to
be
Independence - Disjoint events
Two events, A and B, are said to be
independent , if
Pr( A B)  Pr( A).Pr( B)
Two events are disjoint if
A B
Pr( A B)  0
Example
Consider the following two events in the toss of single
die.
a)
b)
A: observe an odd number
B:observe an even number
C:observe a 1 or 2.
Are A and B independent events?
Are A and C independent events?
Example
Three brands of coffee, X, Y and Z, are to be ranked
according to taste by a judge. Define the following
events:
A: brand X is preferred to Y,
B: brand X is ranked best,
C: brand X is ranked second best
D: brand X is ranked third best.
If the judge actually has no taste preference and thus
randomly assigns ranks to the brands, is event A
independent of events B, C, and D?
Two laws of Probability
Additive law:
The probability of union of two events A and
B is
Pr( A B)  Pr( A)  Pr( B)  Pr( A B)
If A and B are mutually exclusive events,
Pr( A B)  Pr( A)  Pr( B)
Two laws of Probability
Multiplicative law:
The probability of the intersection of two
events A and B is
Pr( A B)  Pr( A).Pr(B A)  Pr(B).Pr( A B)
If A and B are independent,
Pr( A B)  Pr( A).Pr( B)
Two laws of Probability
Multiplicative law:
Prove the probability of the intersection of
any number of events
Bayes’ Rule
Suppose that the events B1, B2 ,......, Bk
form a partition of the space and Pr( Bi )  0
for j  1,....., k.
Then, for every event A in S,
k
Pr( A)   Pr( B j )Pr( A B j )
j 1
Example
Suppose that a person plays a game in which his score
must be one of the 50 numbers 1, 2, …., 50 and
each of these 50 numbers is equally likely to be his
score. The first time he plays the game, his score
is X. He then continues to play the game until he
obtains another score Y such that Y≥X. Assume
that all plays of the game are independent.
Determine the probability of the event A that Y=50.
Example
An electronic fuse is produced by five production lines in a
manufacturing operation. The fuse are costly, are quite reliable,
and are shipped to suppliers in 100-unit lots. Because testing is
destructive, most buyers of the fuses test only a small number
of fuses before deciding to accept or reject lots of incoming
fuses.
All five production lines normally produce only 2% defective fuses,
which are randomly dispersed in the output. Unfortunately,
production line 1 suffered mechanical difficulty 5 % defectives
during the month of March. This situation became known to the
manufacturer after the fuses had been shipped. A customer
received a lot produced in March and tested three fuses. One
failed. What is the probability that the lot was produced on line
1? What is the probability that the lot came from one of the four
other lines?
RANDOM VARIABLE
Random Variable
A real-valued function that is defined on the
space S is called a random variable.
The probability that X takes on the value of x,
Pr(X=x), is defined to be the sum of the
probabilities of all sample points in S
which are assigned the value x by the
function X.
EX: Tossing a coin: A coin is tossed 10 times.
Let X be the number of heads that are
obtained.
Probability distribution


The set of all pairs x, p( x) for which p( X  x)  0
is called the probability distribution for X.
EX: A foreman in manufacturing plant has three
men and three women working for him. He
wants to choose two workers for a special
job. Not wishing to show any biases in his
selection, he decides to select the two
workers at random. Let X denote the number
of women in his selection and find the
probability distribution for X (as histogram)
Binomial distribution
A binomial experiment is one that possesses the
following properties:
1. The experiement consists of n identical trials.
2. Each trial results in one of two outcomes. For lack of
a better nomenclature, we call one outcome a
success, S, and the other a failure, F.
3. The probability of success on a single trial is equal
to p and remains the same from trial to trial. The
probability of failure is equal to 1-p=q.
4. The trials are independent.
5. The random variable of interest is X, the number of
successes observed during the n trials.
Binomial distribution
 n  x n x
p( X  x)    p q
 x
Binomial distribution
Ex1: A coin is tossed ten times. Probability of
observing seven times 1 or 2
Ex2: Consider a family having six children.
Probability of observing two boys