180A HW 1 Solutions
October 4, 2006
1)
A coin is tossed four times resulting in a sequence of heads and tails. Let
A: exactly two heads
B: heads and tails alternate
C: ﬁrst two tosses are headed.
(a) Which events, if any, are mutually exclusive?
(b) Which events, if any, are subsets of other sets?
Answer
a)
Events
B
and
C
are mutually exclusive.
b)
B
⊆
A
.
2)
Compute
P
(
A
c
∪
B
c
)
and
P
(
A
c
∩
(
A
∪
B
))
in terms of
P
(
A
)
,
P
(
B
)
, and
P
(
A
∩
B
)
.
Answer
a)
P
(
A
C
∪
B
C
) = 1

P
(
A
∩
B
)
(by DeMorgan)
b)
P
(
A
C
∪
(
A
∩
B
)) =
P
((
A
C
∪
A
)
∩
(
A
C
∩
B
)) =
P
(
A
C
∩
B
) =
P
(
B
)

P
(
A
∩
B
)
.
3)
Two dice are thrown. Let
A: the sum is odd
B: at least one die lands on an even number
C: the sum is 7
Describe
A
∩
B
,
A
∪
B
,
B
∩
C
,
A
∩
B
c
and
A
∩
B
∩
C
c
and compute the corresponding probabilities.
Answer
We ﬁrst note that
A
⊆
B
, so
A
∩
B
=
A
and
A
∪
B
=
B
. Thus
P
(
A
∩
B
) =
P
(
A
) = 1
/
2
and
P
(
A
∪
B
) =
P
(
B
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 Fall '08
 Castro
 Math, Mutually Exclusive, Ai Aj, State recognized tribes

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