S
ECTION
7.4
I
NDEPENDENT
AND
D
EPENDENT
E
VENTS
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I
NDEPENDENT
E
VENTS
:
2 events that do not affect each other’s outcome
ie: flipping a coin
outcome from each flip is independent
rolling a dice
each roll has a different outcome
drawing cards from a deck with replacement, each card that is
drawn is independent
If 2 events “A” and “B” are independent, then the outcomes of
event “A” do not affect the outcomes of event “B”
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Note: This formula only applies when
A & B are independent events
E
X
: 2 D
ICE
ARE
ROLLED
, G
IVEN
THE
EVENTS
:
A – 1
ST
ROLL
IS
O
DD
B – 2
ND
R
OLL
IS
A
3
F
IND
P(A
AND
B)
The 2 events are independent
One way to check your answer is
to count the number of desired outcomes
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E
X
:
A
BIAS
COIN
IS
FLIPPED
3
TIMES
. P(
HEADS
) = 0.75
W
HAT
IS
THE
PROBABILITY
OF
GETTING
A
T
AILS
, H
EADS
,
AND
THEN
T
AILS
IN
THIS
ORDER
?
The probability of heads is 0.75,
Then the probability of tails is 0.25
Note: Each event is independent
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E
VENT
A –
GETTING
AN
“
A
”
IN
M
ATH
E
VENT
B – G
ETTING
AN
“B”
IN
S
CIENCE
I
F
BOTH
EVENTS
ARE
INDEPENDENT
,
THEN
FIND
THE
FOLLOWING
:
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E
X
:
TWO
SPINNERS
ARE
SPUN
. W
HAT
IS
THE
PROBABILITY
THAT
THE
PRODUCT
WILL
BE
AN
ODD
NUMBER
?
To have an odd product,
both numbers MUST be odd
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B
) W
HAT
IS
THE
PROBABILITY
THAT
THE
PRODUCT
WILL
BE
AN
E
VEN
NUMBER
?
To have an even product, either number could be even
1
st
Method: Add all the cases
where the product is even
2
nd
Method: Subtract the
Complement from 100%
D
EPENDENT
E
VENTS
:
Events “A” and “B” are dependent if the outcome of event “A”
affects the outcome of event “B”
Getting a flat tire and being late for work
Drawing two cards from a deck without replacement
Any experiment where something is removed and not
replaced
Any event that will alter either the “total number of
outcomes” or the “number of desired outcomes”
Create a tree diagram when solving questions involving
dependent events
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E
X
: A
JAR
HAS
3
RED
MARBLES
, 2
BLUE
,
AND
5
GREEN
MARBLES
. W
HAT
IS
THE
PROBABILITY
OF
DRAWING
3
RED
IF
THERE
IS
NO
REPLACEMENT
?
The events are dependent of each other,
since there are no replacements
Use a tree diagram to display the events
1
st
Draw: only two possible outcomes:
Red and non-red marbles
Each time a marble is drawn, there is
one less in the jar
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E
X
: 2
CARDS
ARE
DRAWN
WITHOUT
REPLACEMENT
.
F
IND
THE
PROBABILITY
OF
THE
FOLLOWING
:
A
) 2
HEARTS
ARE
CHOSEN
B
) B
OTH
CARDS
ARE
NOT
HEARTS
C
) O
NLY
ONE
CARD
IS
A
HEART
The events are dependent of each other,
since there are no replacements
Use a tree diagram to display the events
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C
HALLENGE
: T
HERE
ARE
“
X
”
NUMBER
OF
RED
MARBLES
AND
“Y”
NUMBER
OF
GREEN
MARBLES
. I
F
2
MARBLES
ARE
DRAWN
WITHOUT
REPLACEMENTS
,
WHAT
IS
THE
PROBABILITY
OF
GETTING
A
R
ED
FIRST
AND
THEN
A
GREEN
?
Total number of
marbles is
“x + y”
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C
HALLENGE
#2) B
AG
A
CONTAINS
4
GREEN
AND
6
RED
MARBLES
. B
AG
B
CONTAINS
3
GREEN
AND
7
RED
MARBLES
. A
MARBLE
IS
DRAWN
FROM
BAG
A
AND
PLACED
INTO
BAG
B. A
MARBLE
IS
THEN
DRAWN
FROM
BAG
B.
WHAT
IS
THE
PROBABILITY
THAT
A
RED
MARBLE
IS
CHOSEN
?
A marble is drawn from bag A
The number of marbles in bag B
will increase
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H
OMEWORK
:
P429 #2-6, 8, 12 – 14, 16, 22*, 28
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