HSS-CP.B.6 STUDENT NOTES WS #1
1
CONCEPT 1 – Conditional Probability and Dependence
When determining compound probabilities we found the relationship for independent events to be a very
simple multiplication rule, P(A and B) = P(A) • P(B). The occurrences of two independent events in a sequence
don’t alter the probabilities of each other occurring, thus we can multiply their values together. When the
two events are NOT independent then of course they affect each other and their probabilities are altered or
changed by one another. The calculation of a probability that follows a previous event is called a conditional
probability. The phrase GIVEN THAT denotes a conditional situation in which something has already occurred
first. The notation for this relationship is:
P(A|B) this reads the probability of Event A, GIVEN THAT Event B has occurred.
We also learned that if the two events are independent then P (A|B) = P (A) and P (B|A) = P (B). It follows that
if two events are independent then the fact that one of them had already occurred would in no way alter the
probability of the other, thus P (A|B) = P (A).
Determine if the following are Independent or not.
a) What is the probability of
picking the blue marble, GIVEN
THAT the coin was heads?
The flipping of a coin will not
in any way affect the selection
of the marble
c) What is the probability of getting
b) What is the probability of
getting a number greater than 3 on a sum of 10, GIVEN THAT doubles
were rolled?
a die, GIVEN THAT the roll was
even?
P (>3) = 3/6
P (Sum of 10) = 3/36
Independent
P(Blue|Head) = P(Blue)
P (>3|Even) = 2/3
P (Sum of 10|Doubles) = 1/6
Dependent
P(>3|Even) ≠ P(>3)
Dependent
P(Sum of 10|Double) ≠ P(Sum of 10)
d) What is the probability of picking a
red marble on the second pick, GIVEN
THAT on the first pick a red marble
was selected and not replaced?
e) What is the probability of
getting a head on a coin, GIVEN
THAT a head was just previously
flipped on that same coin?
f) What is the probability of rolling
a 4 on a D6, GIVEN THAT a 3 was
just rolled on that same die?
Not Independent
Independent
Independent
In the example below we select a marble, keep it and then select again. This of course will create a dependent
situation because of keeping the selected marble out of the bag.
A bag of marbles has 2 orange, 5 purple and 3 green marbles. Two
marbles are chosen, without replacement. Part of the tree diagram to
this problem is displayed to the right. To calculate the probability of
getting a purple given that an orange had been selected first and not
returned is a conditional probability. One less marble but all 5 purple still
in the bag creates, P(Purple|Orange) = 5/9. In this same relationship the
P(Orange|Orange) = 1/9 because there is one less marble and one less
orange GIVEN THAT orange had already been selected.
P(Purple | Orange)
PURPLE
2
10
Orange
GREEN
ORANGE
HSS-CP.B.6 STUDENT NOTES WS #1
2
A way to make sense of this new relationship is to go back to our Venn diagrams. Let us look at the example
of rolling two dice and summing their values. The sample space for this compound probability is (6)(6) = 36.
P (Sum of 10) = {(4,6), (6,4), (5,5)}
P(Sum of 10) = 3/36
P (Doubles) = {(1,1), (2,2), (3,3), (4,4), (5,5), (6,6)} P(Doubles) = 6/36
Set (Sum of 10) ∩ Set (Doubles) = {(5,5)}
P (Sum of 10 and Doubles) = 1/36
What is the probability of getting The condition is that doubles
a sum of 10 with 2 dice, GIVEN
occurred first. This reduces the
THAT doubles were rolled?
number of possible outcomes
and the intersection is the only
P(Sum of 10|Doubles) = 1/6
values of event A possible.
P(Sum of 10|Doubles) = 1/6 but P (Sum of 10) = 3/36
P(Sum of 10|Doubles) ≠ P(Sum of 10) and so they events are not independent!!
Let us look at another example, what is the probability of rolling a number greater than 3 on a single roll of a
die, given that you know the number was even?
Sample Space = 6
Rolling a value > 3 = {4, 5, 6} P(>3) = 3/6
Rolling an even number = {2, 4, 6}
P(Even) = 3/6
Given that the number was even, the number of outcomes has been reduced
to the size of Set B and the only possible values for Set A to occur are found
in the intersection of Set A and Set B.
2
P (> 3 and Even) 6  2  6  2
P (> 3 | Even) =
= =    =
P(>3|Even) = 2/3
3  6  3  3
P ( Even)
6
P (>3|Even) = 2/3 and P (>3) = 3/6 and so these events are not independent.
What the probability of picking an even
Set Even = {2W, 4W, 8W, 4R, 8R}
number, GIVEN THAT a white was selected? Set White = {1W, 2W, 4W, 5W, 8W, 9W}
P (Even) = 5/10
P (White) = 6/10
Set Even ∩ Set White = {2W, 4W, 8W} P(White & Even) = 3/10
P( E | W ) =
3
 3  10  3
P ( E | W ) = 10 =    =
6  10  6  6
10
P(E) = 5/10
n( E and W ) 3
=
n(W )
6
P(E|W) = 3/6
These events are
independent.
HSS-CP.B.6 STUDENT NOTES WS #1
3
Calculate the conditional probabilities. A bag of marbles contains 2 orange, 5 green and 1 pink.
a) Given that a pink was selected
but not replaced, on the second
pick what is the P (Orange)?
b) Given that an orange was
selected and replaced, on the
second pick what is the P (Green)?
c) Given that a green was selected
and not replaced, on the second
pick what is the P (Green)?
P (Green) = 5/8
P (Green) = 4/7
P (Orange) = 2/7
Complete the Venn diagram and determine the missing values for when one D12 is rolled.
The Universal Set is (1, 2, 3, …, 10, 11, 12)
Set E = Evens
Set E = {2, 4, 6, 8, 10, 12}
P(E) = 6/12
Set T = #’s > 4
Set T = {5, 6, 7, 8, 9, 10, 11, 12}
P(T) = 8/12
Set G = First 2 & Last 2
Set G = {1, 2, 11, 12}
P(G) = 4/12
Set F = Factors of 12
Set F = {1, 2, 3, 4, 6, 12}
P(F) = 6/12
a) Set E ∩ Set G? ___________
{2, 12}
b) Set E ∩ Set T? ___________
{6, 8, 10, 12}
P(E and G) = ________
2/12
P(E and T) = ________
4/12
P(E|G) = _______
2/4
P(T|E) = _______
4/6
c) Set F ∩ Set G? ___________
{1, 2, 12}
d) Set G ∩ Set T? __________
{11, 12}
P(F and G) = ________
3/12
P(G and T) = ________
2/12
P(F|G) = _______
3/4
P(G|T) = _______
2/8
Complete the Venn diagram and determine the probability and if they are independent or not.
a) Given a 20 sided dice.
P(value greater than 15) = _________
5/20
P(factor of 20) = _________
6/20
Set (>15) ∩ Set (Factor of 20) = ______________
{20}
P (>15 and Factor of 20) = ________
1/20
What is P (Factor of 20|>15)? = _________
1/5
What is the P (>15|Factor of 20)? = _________
1/6
Independent? No
HSS-CP.B.6 STUDENT NOTES WS #1
4
CONCEPT 2 – Conditional Probability and Mutually Exclusive
The relationship for determining a conditional probability is P ( A | B ) =
n( A and B )
but if the two events have
n( B )
no elements in common, then they have no intersection. Thus n(A and B) is zero and P(A|B) = 0.
Our Venn diagram helps us make sense of this because if event R (Red Marbles) has occurred but event W
(White Marbles) has NO elements in event R, the it is IMPOSSIBLE FOR IT TO OCCUR, P(W|R) = 0.
If Set W (White Marbles) and Set R (Red
Marbles) are mutually exclusive, they have
no intersection and if event R occurs it
would be impossible for event W to occur.
n(W and R ) 0
P (W | R ) =
= =0
n( R )
4
Name: _____________________________ Period ______
HSS-CP.B.6 WORKSHEET #1
1. What do we mean by a conditional probability?
2. Translate this notation into English, P (A|B).
3. Calculate the conditional probabilities. A bag of marbles contains 3 red, 5 blue, and 2 green.
a) Given that a green was selected and replaced,
on the second pick what is the
b) Given that a red was selected and replaced,
on the second pick what is the
P(Blue|Green) = _________
P(Red|Red) = _________
c) Given that a blue was selected and kept,
on the second pick what is the
d) Given that a green was selected and kept,
on the second pick what is the
P(Red|Blue) = _________
P(Green|Green) = _________
4. Complete the Venn diagram and determine the missing values for when one die is rolled.
Set E = Evens
Set E = {2, 4, 6}
P(E) = 3/6
Set L = #’s < 3
Set L = {1, 2}
P(L) = 2/6
Set M = Middle
Set M = {3}
P(M) = 1/6
Set F = Factors of 6
Set F = {1, 2, 3, 6}
P(F) = 4/6
a) Set E ∩ Set L? ___________
b) Set E ∩ Set L? ___________
P(E and L) = ________
P(E and L) = ________
P(E|L) = _______
P(L|E) = _______
c) Set E ∩ Set F? ___________
d) Set M ∩ Set L? __________
P(E and F) = ________
P(M and L) = ________
P(E|F) = _______
P(M|L) = _______
e) Set M ∩ Set F? __________
f) Set L ∩ Set F? __________
P(M and F) = ________
P(L and F) = ________
P(M|F) = _______
P(L|F) = _______
1
2
HSS-CP.B.6 WORKSHEET #1
5. Complete the Venn diagram and determine the probability and if they are independent or not.
a) Given a 6 sided dice.
P(A) = P(value greater than 2) = _________
P(B) = P(even roll) = _________
Set A ∩ Set B = ______________
P(A and B) = ______
What is P(B|A)? = _________
What is the P(A|B)? = _________ Independent? Yes or No
b) Given a spinner has 4 equal color quadrants (red, blue, green &
orange) and a die with 6 sides.
P(A) = P(getting blue) = _________
P(B) = P(a multiple of 3) = _________
Set A ∩ Set B = ______________
P(A and B) = ______
What is P(B|A)? = _________
What is the P(A|B)? = _________
Independent? Yes or No
c) Given two six sided dice.
P(A) = P(sum less than 5) = _________
P(B) = P(doubles) = _________
Set A ∩ Set B = ______________
P(A and B) = ______
What is P(B|A)? = _________
What is the P(A|B)? = _________ Independent? Yes or No
d) Given the roll of a single die.
P(A) = P(odd number less than 5) = _________
P(B) = P(numbers less than 3) = _________
Set A ∩ Set B = ______________
P(A and B) = ______
What is P(B|A)? = _________
What is the P(A|B)? = _________ Independent? Yes or No
d) Given the roll of a single die.
Set A = {1, 5, 6}
P(A) = _________
Set B = {3 4, 5, 6}
P(B) = _________
Set A ∩ Set B = ______________
P(A and B) = ______
What is P(B|A)? = _________
What is the P(A|B)? = _________ Independent? Yes or No
3
HSS-CP.B.6 WORKSHEET #1
6. Using the Venn diagram explain why P ( A | B ) =
n( A & B )
n( B )
7. Calculate the conditional probabilities.
a) A coin is flipped and then a die (D6) is rolled.
Given that a head was flipped,
b) Two dice (D6) are rolled. Given that doubles were
rolled with 2 dice, what is P(sum<3|doubles)?
what is the P(rolling a 5|head) = _____________.
what is P(sum<3|doubles) = __________.
c) A bag of marbles with 3 green, 5 red and 2 yellow.
Given that a yellow was selected and kept, on the
second pick,
d) A card is being selected from a standard deck.
Given that a red card was selected,
what is the P(red|yellow) = ____________.
what is the P(face card|red card) = ___________.
e) A card is being selected from a standard deck.
Given that a face card was selected,
f) A single six sided die is rolled. Given that it came up
even,
what is the P(queen|face card) = ___________.
what is the P(value<2|even) = ____________.
8. In a regular deck of cards,
a) P (red jack | red card) =
________
b) P (face card | red card) = ________
c) P (face card | diamond) = ________
d) P (numerical card | black card) = ________
e) P (ace | face card) =
f) P (num. card < 5 | numerical card) = _________
________
,/
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Name:
y*^
Period
1.Whatdowemeanbyaconditionatprobabitity?
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2. Translate this notation
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Phv-a-Ee '(Gn utao#l'
ir
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la)
into English,
P (A I B).
3. Calculate the conditional probabilities. A bag of marbles contains 3 red, 5 blue, and 2 green.
that a green was selected and replaced.
on the second pick what is the
a) Given
=
P(BIuelGreen)
b) Given that a red was selected and replaced,
on the second pick what is the
P(Red I Red) =
TO
d) Given that a green was selected and kept,
on the second pick what is the
keet,
c) Given that a blue was selected and,.on the second pick what is the
-a
A
P(Red lBlue)
= A
P(Green lGreen) =
_
4. Complete the Venn diagram and determine the missing values for when one die is rotled.
= Evens
Set
E
= {2,4,61
P(E) = 376
SetL=#'s<3
5s1
l_
= {1, 2}
P(Ll = 216
Set M = Middle
Set M = {3}
P(M)=
SetF={L,2,3,61
Pffl =
Set
Set
E
F
= Factors of 6
a) Set E
n
Set L?
1"3
U
and L)
P(ElL)
=I
c)set
E
P(E
n
n
Set L?
U
td|.
P(E
,*€
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tct 1l
,*l
tdt
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and L) =
P(LlE)=
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and F) =
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a15
b) Set E
JdE
P(E
176
rl ?\
L3
U
d) Set
M
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q \e
)zl
1-
-b
n
Set
L?
t
P(M and L)=
4
?-
P(E
lF)
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P(M lL) =
e)setMnSetrl l:(
P(M and F)=
P(M lF)
=+
+
t,
t*rl /
\
J.rl
3\ bL i
f) Set L n Set
P(L
and F) =
P(LlF)=
F?
U
H SS.CP,
8.6 WO RKSH EET # 7
they are ildependent o
5. Complete the Venn diagram and determine the probability and if
F
a) Given a 5 sided dice.
grrrg
SetAn
P(A and B)
SetB=
what is P(B I nlt
:r
Z
=b
/ir
What is the P(AIB)? =',' a)
= i,3)'
\J-
lndependent?
@ or
No
or
No
green &
b) Given a spinner has 4 equal color quadrants (red, blue,
orange) and a die with 6 sides.
P(A) = P(getting blue)=
P(A and B) =
SetAn SetB=
lndependent? Yes
What is the P(AlB)? =
What is P(B|A)? =
P(A) = P(sum less than 5l =
P(B) =P(doubles)
k=
-i-ib-'' -/-i
P(B
I
N?
+ Vt" Yvee
)
,1,,1\
= &
) /t,,),(2,$
SetAn SetB= (t
What is
U
dice.
c) Given two six sided
=
P(A and B\
%
What is
=
,
3A
the P(AlB)?
,/ L\
;\ T I
\7-
lndependent? Yes
"t@
d) Given the roll of a single die.
P(A) = P(odd number less than 5) =
P(B) = P(numbers less
than 3)
=
P(A and B) =
SetAn SetB=
d) Given the roll of a single
SetA=i1,5,6)
What is the P(AlB)? =
ai7,'?'-
\!'r\\\
t".
P(A)= *-3,'
,,Lr1
set B = t3 4,s,61 ,(tl. Jg-*_
setAn
setB=
What is P(BlA)?
1:,r\)r'fino
=
What
Br=
is
lndependent? Yes
or No
tndependentfre)
or No
$o-"-"-
2_ \ \
?.
the P(AlB)?
=
{6tb
t/
II
f
Hss-cP.B.6 WORKSHEET #7
G. Using
the venn diagram exptain why p(Al
U
a)=443-B)
ia*
+F G 6!
.* Event,
\A
&i
7.
Calculate the conditional'probabilities.
a) A coin is flipped and then a die (DG) is rolled.
Given that a head was flipped,
what
is
the P(rolling a 5lhead)
=
T:
what is P(sum<3ldoubles) =
c) A bag of marbles with 3 green, 5 red and 2 yellow.
Given that a yellow was selected and kqpl, on the
second pick,
what
is
the P(red lyellow)
q5
=
\
tX- Ftc"4-
d
7(Q ?{ace-) = t'
z(Forn\
what
is
the P(queen lface card)
3tL
=
t2.
e) P (ace I face card) =
what is the P(value<2leven) =
n
LL
I red card)
(
l2-
-->_
=
c) P (face card I diamond) =
o
even,
C""A>
8. ln a regular deck of cards,
a) P (red jack
f) A single six sided die is rolled. Given that it came up
t), qu*q
=+
2*
d) A card is being selected from a standard deck.
Given that a red card was selected,
what is the P(face card lred card) =
e) A card is being selected from a standard deck.
Given that a face card was selected,
-f
b)Two dice (D6) are rolled. Given that doubles were
rolled with 2 dice, what is p(sum<3ldoubles)?
Aa"
2t3
%
b) P (face card I red card) =
d) P (numerical card I black card) =
f)
P (num. card <
5 | numerical card) =