Let’s be a set of outcomes and let A and B be events with outcomes in S．Let-B de

游客2024-01-13 56

问题 Let’s be a set of outcomes and let A and B be events with outcomes in S．Let-B denote the set of all outcomes in S that are not in B and let P(A)denote the probability that event A occurs．What is the value of P(A)?
(1) P(A ∪ B)=0．7
(2) P(A ∪ -B)=0．9

选项 A、Statement(1) ALONE is sufficient，but statement (2) alone is not sufficient．
B、Statement(2) ALONE is sufficient，but statement (1) alone is not sufficient．
C、BOTH statements TOGETHER are sufficient，but NEITHER statement ALONE is sufficient．
D、EACH statement ALONE is sufficient．
E、Statements(1) and(2) TOGETHER are NOT sufficient．

答案 C

解析 The general addition rule for sets applied to probability gives the basic probability equation
P(A∪B) = P(A) + P(B) -P(A∩ B).
(1) Given that P(A ∪ B) = 0.7, it is not possible to determine the value of P(A) because nothing is known about the relation of event A to event B. For example, if every outcome in event B is an outcome in event A, then A ∪ B = A and we have P{A ∪ B) = P(A) = 0.7. However, if events A and B are mutually exclusive (i.e., P(A ∩ B) = 0) and P{B} = 0.2, then the basic probability equation above becomes 0.7 = P(A) + 0.2 - 0, and we have P{A} = 0.5; NOT sufficient.
(2) Given that P(A ∪ ~B) = 0.9, it is not possible to determine the value of P{A} because nothing is known about the relation of event A to event ~B. For example, as indicated in the first figure below, if every outcome in event ~B is an outcome in event A, then A ∪ ~B = A and we have P(A ∪ ~B) = P(A) = 0.9. However, as indicated in the second figure below, if events A and ~B are mutually exclusive (i.e., P(A ∩ ~B) = 0) and P(~B) = 0.2, then the basic probability equation above, with ~B in place of 5, becomes 0.9 = P{A} + 0.2 - 0, and we have P(A) = 0.7; NOT sufficient.

Given (1) and (2), if we can express events as a union or intersection of events A ∪ B and A ∪ ~B, then the basic probability equation above can be used to determine the value of P{A}.The figure below shows Venn diagram representations of events A ∪ B and A ∪ ~B by the shading of appropriate regions.

Inspection of the figure shows that the only portion shaded in both Venn diagrams is the region representing event A. Thus, A is equal to the intersection of A ∪ B and A ∪ ~B, and hence we can apply the basic probability equation with event A ∪ B in place of event A and event A ∪ ~B in place of event B. That is, we can apply the equation
P(C∪D) = P{C} + P(D) -P(C∩D)
with C = A ∪ B and D = A ∪ ~B. We first note that P(C) = 0.7 from (1), P(D) = 0.9 from (2), and P(C ∩D) = P(A). As for P(C ∪ D), inspection of the figure above shows that C ∪ D encompasses all possible outcomes, and thus P(C ∪ D) = 1. Therefore, the equation above involving events C and D becomes 1 = 0.7 + 0.9 - P(A), and hence P(A) = 0.6.
The correct answer is C;
both statements together are sufficient.

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