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Sec 4.4
The multiplication Rule and
conditional probability
Independent events
 Two events A and B are independent events
if the fact that A occurs does NOT effect the
probability of B occurring.
Multiplication Rule 1
 If two events are independent, the probability
of both happening is
P(A and B)= P(A) • P(B)
Examples
 A coin is flipped and a die is rolled at the same
time. What is the probability of getting a tail and a
4?
A card is drawn at random and then replaced.
What is the probability of drawing a queen first
and then an ace?
More examples
 A box contains 3 red balls, 2 blue balls and 5
green balls. A ball is selected, replaced and a
second ball is selected.

Write the sample space.

Write the probability of each event.
Refer to the last example and answer the
following probabilities:
a) Selecting 2 blue balls.
b) Selecting 1 blue ball and then 1 green ball.
c) Selecting 1 red ball and then 1 blue.
Color blindness
 Studies show that approximately 9% of men
have color blindness. If 3 males are selected
at random what is the probability that none of
them have color blindness?
Refer to the color blindness
problem
 If 3 men are chosen at random. What is the
sample space for color blindness?
 Do you think each event has the same
probability of happening?
Dependent events
 When the outcome of the first event affects
the outcome of the second event in such a
way that the probability is changed, the
events are dependent.
Multiplication Rule #2
 When two events are dependent, the
probability of both happening is
 P(A and B)= P(A) · P(B|A)
Example 4-28
 A person owns a collection of 30 CDs, of
which 5 are country music. If 2 CDs are
selected at random, find the probability that
both are country music.
Example 4-29
 The World Wide Insurance Company found that
53% of the residents of a city had homeowner’s
insurance (H) with the company. Of these clients,
27% also had automobile insurance (A) with the
company. If a resident is selected at random, find
the probability that the resident has both
homeowner’s and automobile insurance with the
World Wide Insurance Company.
Example 4-30
 Three cards are drawn from an ordinary deck and
not replaced. Find the probability of these.
 A.) Getting 3 jacks.
 B.) Getting an ace, a king, and a queen in order.
 C.) Getting a club, a spade, and a heart in order.
 D.) Getting 3 clubs.
Example 4-31
 Box 1 contains 2 red balls and 1 blue ball.
Box 2 contains 3 blue balls and 1 red ball. A
coin is tossed. If it falls heads up, box 1 is
selected and a ball is drawn. If it falls tails
up, box 2 is selected and a ball is drawn. Find
the probability of selecting a red ball.
Formula for Conditional
Probability
 The probability that the second event B
occurs given that the first event A has
occurred can be found by dividing the
probability that both events occurred by the
probability that the first event has occurred.
The formula is:
p( AandB)
p( B A) 
p( A)
Example 4-32
 A box contains black chips and white chips. A
person selects two chips without replacement. If the
probability of selecting a black chip and a white
chip is 15/56, and the probability of selecting a
black chip on the first draw is 3/8, find the
probability of selecting the white chip on the
second draw, given that the first chip selected was a
black chip.
Example 4-33
 The probability that Sam parks in a noparking zone and gets a parking ticket is
0.06, and the probability that Sam cannot
find a legal parking space and has to park in
the no-parking zone is 0.20. On Tuesday,
Sam arrives at school and has to park in a nopark zone. Find the probability that he will
get a parking ticket.
Example 4-34
 A recent survey asked 100 people if they thought
women in the armed forces should be permitted to
participate in combat. The results of the survey are
shown.
 Male Yes-32, No-18 Total-50
 Female Yes-8 No-42 Total-50
 Find these probabilities.
 A.) The respondent answered yes, given that the
respondent was a female.
 B.) The respondent was a male, given that the
respondent answered no.
Probabilities for “At Least”
 The multiplication rules can be used with the
complementary event rule (Section 4-2) to
simplify solving probability problems
involving “at least.” Examples 4-35, 4-36,
and 4-37 illustrate how this is done.
Example 4-35
 A game is played by drawing four cards from
an ordinary deck and replacing each card
after it is drawn. Find the probability of
winning if at least one ace is drawn.
Example 4-36
 A coin is tossed 5 times. Find the probability
of getting at least one tail.
Example 4-37
 The Newckware Association of America
reported that 3% of ties sold in the United
States are bow ties. If 4 customers who
purchased a tie are randomly selected, find
the probability that at least one purchased a
bow tie.
Homework
Sec 4.4 page 209-212
#1-51 Every other odds. i.e 1,5,9,
etc.