Lesson 14-3
Probability of
Compound Events
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Objectives
• Find the probability of two independent events
or dependent events
• Find the probability of two mutually exclusive
or inclusive events
Vocabulary
•
•
•
•
•
•
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Simple event –
Compound event –
Independent events –
Dependent events –
Complements –
Mutually exclusive –
Inclusive –
Probability Part 1
• Independent Events –
one event does not affect the outcome
of another
P(A and B) = P(A)P(B)
A
B
Example: Rolling a 5 on a die twice or
flipping a coin to get a head twice
• Dependent Events –
one event effects the outcome of
another
P(5 and 5) = P(5)P(5)
= (1/6)(1/6)
= (1/36)
P(A and B) = P(A)P(B given A)
Example: Drawing two cards from a
P(A and A) = P(A)P(A given 1st A)
normal deck of cards without
= (4/52)(3/51)
replacing the first card and getting two
= (12/2652)
aces
Probability Part 2
• Mutually Exclusive Events –
events that cannot occur at the same time
P(A or B) = P(A)+P(B)
Example: Drawing a heart or a club from a
normal deck of cards
A
• Inclusive Events –
events overlap
P(A or B) = P(A)+P(B) - P(A and B)
Example: Drawing an ace or a heart from a
normal deck of cards
• Complement Events –
all other events
Example: flipping a coin to get a head or
not a head (a tail)
B
P(H or C) = P(H)+P(C)
=(1/4)+(1/4)
= (1/2)
A
B
P(A or H) = P(A)+P(H) – P(A and H)
=(4/52)+(13/52) - (1/52)
= (16/52)
P(head) + P(not a head) = 1
Example 1
Roberta is flying from
Birmingham to Chicago
to visit her grandmother.
She has to fly from Birmingham to Houston on the
first leg of her trip. In Houston she changes planes
and heads on to Chicago. The airline reports that
the flight from Birmingham to Houston has a 90%
on time record, and the flight from Houston to
Chicago has a 50% on time record. What is the
probability that both flights will be on time?
Example 1 cont
Definition of
independent events
0.9
0.5
Multiply.
Answer: The probability that both flights will be on-time is 45%.
Example 2
At the school carnival, winners in the ring-toss game are
randomly given a prize from a bag that contains 4
sunglasses, 6 hairbrushes, and 5 key chains. Three prizes
are randomly drawn from the bag and not replaced.
A. Find P(sunglasses, hairbrush, key chain).
The selection of the first prize affects the selection of the next
prize since there is one less prize from which to choose. So, the
events are dependent.
First prize:
Second prize:
Third prize:
Example 2 cont
Substitution
Multiply.
Answer: The probability of drawing sunglasses,
a hairbrush, and a key chain is 4/91 ≈ 0.044
Example 2b cont
At the school carnival, winners in the ring-toss game are
randomly given a prize from a bag that contains 4
sunglasses, 6 hairbrushes, and 5 key chains. Three prizes
are randomly drawn from the bag and not replaced.
B. Find P(hairbrush, hairbrush, key chain).
Notice that after selecting a hairbrush, not only is there one fewer
prize from which to choose, there is also one fewer hairbrush.
Multiply.
Answer: The probability of drawing two hairbrushes and
then a key chain is 5 out of 91 or 5/91 ≈ 0.055
Example 2c cont
At the school carnival, winners in the ring-toss game are
randomly given a prize from a bag that contains 4
sunglasses, 6 hairbrushes, and 5 key chains. Three prizes
are randomly drawn from the bag and not replaced.
C. Find P(sunglasses, hairbrush, not key chain).
Since the prize that is not a key chain is selected after the first
two prizes, there are 10 – 2 or 8 prizes that are not key chains.
Multiply.
Answer: The probability of drawing sunglasses, a
hairbrush, and not a key chain is 32/455 ≈ 0.07
Example 3
Alfred is going to the Lakeshore Animal Shelter to pick a
new pet. Today, the shelter has 8 dogs, 7 cats, and 5
rabbits available for adoption. If Alfred randomly picks an
animal to adopt, what is the probability that the animal
would be a cat or a dog?
Since a pet cannot be both a dog and a cat, the events are
mutually exclusive.
Example 3 cont
Definition of mutually
exclusive events
Substitution
Add.
Answer: The probability of randomly picking a cat or
a dog is ¾ or 0.75
Example 4
A dog has just given birth to a litter of 9 puppies. There
are 3 brown females, 2 brown males, 1 mixed-color
female, and 3 mixed-color males. If you choose a puppy
at random from the litter, what is the probability that the
puppy will be male or mixed-color?
Since three of the puppies are both mixed-colored and males,
these events are inclusive.
Definition of
inclusive events
Example 4 cont
Substitution
LCD is 9.
Simplify.
Answer: The probability of a puppy being a male or
mixed-color is 2/3 or about 67%.
Summary & Homework
• Summary:
– For independent events,
use P(A and B) = P(A)  P(B)
– For dependent events,
use P(A and B) = P(A)  P(B following A)
– For mutually exclusive events,
use P(A or B) = P(A) + P(B)
– For inclusive events,
use P(A or B) = P(A) + P(B) – P(A and B)
• Homework:
– none