题目信息
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) ?
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.
参考答案及共享解析
共享解析来源为网络权威资源、GMAT高分考生等； 如有疑问，欢迎在评论区提问与讨论
本题耗时：
已选答案：
正确答案：
C:BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
Arithmetic Probability; Sets
The general addition rule for sets applied to probability gives the basic probability equation
P(A
B) = P(A) + P(B) – P(A 
B).
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.
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 B, 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 event A 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.
The general addition rule for sets applied to probability gives the basic probability equation
P(A
B) = P(A) + P(B) – P(A B).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.
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 B, 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 event A 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 equationP(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.
加入收藏
在线答疑
题目来源
