     
negation: P ( E ) + P ( E  ) = 1
or: P ( A  B ) = P ( A ) + P ( B ) - P ( A  B )
P(A B )
if: P ( A / B ) =
P(B)
and: P ( A  B ) = P ( A ) P ( B / A )
     
negation: P ( E ) + P ( E  ) = 1
Roll 3 dice. What is the
probability that AT LEAST
two land on the same number?
next
negation: P ( E ) + P ( E  ) = 1
Roll 3 dice. What is the
probability that AT LEAST
two land on the same number?
S = event that at least 2 same
S  = event that all are different
If you do NOT have
at least 2 landing on the
same number then
they all land on different
numbers.
next
negation: P ( E ) + P ( E  ) = 1
Roll 3 dice. What is the
probability that AT LEAST
two land on the same number?
S = event that at least 2 same
S  = event that all are different
P(S ) =
6x5x4
6x6x6
20
=
36
16
P(S) =
36
next
negation: P ( E ) + P ( E  ) = 1
Roll 3 dice. What is the
probability that AT LEAST
two land on the same number?
Draw 3 marbles
without replacement.
What is the probability
that AT LEAST one is blue?
S = event that at least 2 same
S  = event that all are different
P(S ) =
6x5x4
6x6x6
20
=
36
16
P(S) =
36
next
negation: P ( E ) + P ( E  ) = 1
Roll 3 dice. What is the
probability that AT LEAST
two land on the same number?
S = event that at least 2 same
Draw 3 marbles
without replacement.
What is the probability
that AT LEAST one is blue?
B = event that at least 1 is blue
S  = event that all are different B  = event that all are red
P(S ) =
6x5x4
6x6x6
16
P(S) =
36
20
=
36
If you do NOT have
at least one blue then
they are all red.
next
negation: P ( E ) + P ( E  ) = 1
Roll 3 dice. What is the
probability that AT LEAST
two land on the same number?
S = event that at least 2 same
Draw 3 marbles
without replacement.
What is the probability
that AT LEAST one is blue?
B = event that at least 1 is blue
S  = event that all are different B  = event that all are red
P(S ) =
6x5x4
6x6x6
16
P(S) =
36
20
=
36
P(B ) =
5C3
10
=
56
8C3
46
P(B) =
56
next
negation: P ( E ) + P ( E  ) = 1
Roll 3 dice. What is the
probability that AT LEAST
two land on the same number?
S = event that at least 2 same
Draw 3 marbles
without replacement.
What is the probability
that AT LEAST one is blue?
B = event that at least 1 is blue
S  = event that all are different B  = event that all are red
P(S ) =
6x5x4
6x6x6
16
P(S) =
36
20
=
36
P(B ) =
5C3
8C3
10
=
56
46
P(B) =
56
return to outline
or: P ( A  B ) = P ( A ) + P ( B ) - P ( A  B )












K
Q
J
K
Q
J
K
Q
J
K
Q
J
Draw 2 cards without replacement from the deck pictured above.
What is the probability that they are both red or both kings?
next
or: P ( A  B ) = P ( A ) + P ( B ) - P ( A  B )












K
Q
J
K
Q
J
K
Q
J
K
Q
J
Draw 2 cards without replacement from the deck pictured above.
What is the probability that they are both red or both kings?
R = both red
6C2
15
=
P(R) =
66
12C2
6C2
= 6x5
2x1
12C2
= 12 x 11
2x1
next
= 15
= 66
or: P ( A  B ) = P ( A ) + P ( B ) - P ( A  B )












K
Q
J
K
Q
J
K
Q
J
K
Q
J
Draw 2 cards without replacement from the deck pictured above.
What is the probability that they are both red or both kings?
K = both kings
R = both red
P(R) =
6C2
12C2
15
=
66
P(K) =
4C2
12C2
6
=
66
next
or: P ( A  B ) = P ( A ) + P ( B ) - P ( A  B )












K
Q
J
K
Q
J
K
Q
J
K
Q
J
Draw 2 cards without replacement from the deck pictured above.
What is the probability that they are both red or both kings?
K = both kings
R = both red
P(R) =
6C2
12C2
15
=
66
P(K) =
4C2
6
=
66
12C2
R K=

K K
one hand
There is only 1 pair of cards satisfying:
They are both red AND they are both kings.
next
P(R K)=
1
66
or: P ( A  B ) = P ( A ) + P ( B ) - P ( A  B )












K
Q
J
K
Q
J
K
Q
J
K
Q
J
Draw 2 cards without replacement from the deck pictured above.
What is the probability that they are both red or both kings?
K = both kings
R = both red
P(R) =
6C2
12C2
15
=
66
P(K) =
4C2
12C2
6
=
66
P(R K)= P(R) + P(K) - P(R K)
15
6
1
+
=
66
66
66
next
R K=

K K
one hand
P(R K)=
1
66
or: P ( A  B ) = P ( A ) + P ( B ) - P ( A  B )












K
Q
J
K
Q
J
K
Q
J
K
Q
J
Draw 2 cards without replacement from the deck pictured above.
What is the probability that they are both red or both kings?
K = both kings
R = both red
P(R) =
6C2
12C2
15
=
66
P(K) =
4C2
12C2
6
=
66
P(R K)= P(R) + P(K) - P(R K)
15
6
1
20
+
=
66
66
66
66
return to outline
R K=

K K
one hand
P(R K)=
1
66
P(A B )
if: P ( A / B ) =
P(B)
In a certain population, 13% are left handed males
and 52% are male. A person is selected at random from this
population. What is the probability he is left handed IF he is male?
P(LM)
if: P ( L / M ) =
P(M)
=
next
P(A B )
if: P ( A / B ) =
P(B)
In a certain population, 13% are left handed males
and 52% are male. A person is selected at random from this
population. What is the probability he is left handed IF he is male?
P(LM)
if: P ( L / M ) =
P(M)
=
.13
.52
= .25
next
P(A B )
if: P ( A / B ) =
P(B)
In a certain population, 13% are left handed males
and 52% are male. A person is selected at random from this
population. What is the probability he is left handed IF he is male?
P(LM)
if: P ( L / M ) =
P(M)
=
.13
.52
= .25
return to outline
and: P ( A  B ) = P ( A ) P ( B / A )












K
Q
J
K
Q
J
K
Q
J
K
Q
J
Draw 2 cards without replacement from the deck pictured above.
What is the probability that
the first is a heart and the second is a heart?
next
and: P ( A  B ) = P ( A ) P ( B / A )












K
Q
J
K
Q
J
K
Q
J
K
Q
J
Draw 2 cards without replacement from the deck pictured above.
What is the probability that
the first is a heart and the second is a heart?
and: P ( R1  R2 ) = P (R1 ) P (R2 / R1 )
next
and: P ( A  B ) = P ( A ) P ( B / A )












K
Q
J
K
Q
J
K
Q
J
K
Q
J
Draw 2 cards without replacement from the deck pictured above.
What is the probability that
the first is a heart and the second is a heart?
and: P ( R1  R2 ) = P (R1 ) P (R2 / R1 )
3 hearts
12 cards
3
12
next
and: P ( A  B ) = P ( A ) P ( B / A )












Q
J
K
Q
J
K
Q
J
K
Q
J
K
Draw 2 cards without replacement from the deck pictured above.
What is the probability that
the first is a heart and the second is a heart?
and: P ( R1  R2 ) = P (R1 ) P (R2 / R1 )
If the first card is a heart:
3 hearts
2 remaining hearts
12 cards
11 remaining cards
3
12
2
11
next
and: P ( A  B ) = P ( A ) P ( B / A )
 










K
J
K
Q
J
K
Q
J
K
Q
J
Q
Draw 2 cards without replacement from the deck pictured above.
What is the probability that
the first is a heart and the second is a heart?
and: P ( R1  R2 ) = P (R1 ) P (R2 / R1 )
3 hearts
2 remaining hearts
12 cards
11 remaining cards
3
12
3
2
=
66
11
return to outline