The event A ∩ B can be written as the union of two disjoint events as follows:
A ∩ B = (A ∩ B ∩ C) ∪ (A ∩ B ∩ C
c
),
so that
P(A ∩ B) = P(A ∩ B ∩ C) + P(A ∩ B ∩ C
c
). (2)
Similarly,
P(A ∩ C) = P(A ∩ B ∩ C) + P(A ∩ B
c
∩ C). (3)
Combining Eqs. (1)-(3), we obtain the desired result.
Solution to Problem 1.10. Since the events A ∩ B
c
and A
c
∩ B are disjoint, we
have using the additivity axiom repeatedly,
P
(A∩B
c
)∪(A
c
∩B)
= P(A∩B
c
)+P(A
c
∩B) = P(A)−P(A∩B)+P(B)−P(A∩B).
Solution to Problem 1.14. (a) Each possible outcome has probability 1/36. There
are 6 possible outcomes that are doubles, so the probability of doubles is 6/36 = 1/6.
(b) The conditioning event (sum is 4 or less) consists of the 6 outcomes
(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (3, 1)
,
2 of which are doubles, so the conditional probability of doubles is 2/6 = 1/3.
(c) There are 11 possible outcomes with at least one 6, namely, (6, 6), (6, i), and (i, 6),
for i = 1, 2, . . . , 5. Thus, the probability that at least one die is a 6 is 11/36.
(d) There are 30 possible outcomes where the dice land on different numbers. Out of
these, there are 10 outcomes in which at least one of the rolls is a 6. Thus, the desired
conditional probability is 10/30 = 1/3.
Solution to Problem 1.15. Let A be the event that the first toss is a head and
let B be the event that the second toss is a head. We must compare the conditional
probabilities P(A ∩ B |A) and P(A ∩ B |A ∪ B). We have
P(A ∩ B |A) =
P
(A ∩ B) ∩ A
P(A)
=
P(A ∩ B)
P(A)
,
and
P(A ∩ B |A ∪ B) =
P
(A ∩ B) ∩ (A ∪ B)
P(A ∪ B)
=
P(A ∩ B)
P(A ∪ B)
.
Since P(A ∪ B) ≥ P(A), the first conditional probability above is at least as large, so
Alice is right, regardless of whether the coin is fair or not. In the case where the coin
is fair, that is, if all four outcomes HH, HT , T H, T T are equally likely, we have
P(A ∩ B)
P(A)
=
1/4
1/2
=
1
2
,
P(A ∩ B)
P(A ∪ B)
=
1/4
3/4
=
1
3
.
A generalization of Alice’s reasoning is that if A, B, and C are events such that
B ⊂ C and A ∩ B = A ∩ C (for example, if A ⊂ B ⊂ C), then the event A is at least
4
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qq_345887822018-01-09很好啊很好啊
