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I have the following homework assignment that I've already finished, but am confused on whether I've gotten right/wrong, and was hoping someone could help explain so I understand the problem better.

An oil exploration company currently has two active projects, one in Asia and the other in Europe. Let A be the event that the Asian projet is successful and B be the event that the European project is successful. Suppose that A and B are independent events with P(A)=0.4 and P(B)=0.7

a.) If the Asian project is not successful, what is the probability that the European project is also not successful? Explain your reasoning.

b.) What is the probability that at least one of the two projects will be successful?

c.) Given that at least one of the two projects is successful, what is the probability that only the Asian project is successful?"

Here is what I've gotten for each part:

$$P(A \cap B) = P(A)P(B) = (.4)(.7) = .28 $$

a.) $P(B^c) = 1 - 0.7 = 0.3 $

b.) $P(A \cup B) = P(A) + P(B) - P(A \cap B) = .4 + .7 - .28 = .82$

c.) $P(A) - P(A \cap B) = .4 - .28 = .12 $ $.12/.82 = .146$

I am confused in that the two events are independent of each other and the book states that for part a the answer should be .126 instead of what I got. Am I doing these problems correctly or am I committing some error?

Did
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Valrok
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  • Part a) is correctly done. – André Nicolas Feb 21 '14 at 17:32
  • So do I need to get $P(B^c | A^c)$ or not for part a? I originally thought I was correct and there may be a chance the book is wrong, but I don't see how to get 0.126 as the answer for part a – Valrok Feb 21 '14 at 17:35
  • You are right, $B^c$ and $A^c$ are independent. But if you really really feel like it, you can calculate. We have $\Pr(B^c|A^c)=\frac{\Pr(B^c\cap A^c)}{\Pr(A^c)}$. After some grinding you will get $0.3$. – André Nicolas Feb 21 '14 at 17:41
  • Ah, I guess I may have stumbled into one of those few times that the book is incorrect with part a. For the other parts my answers were the same as the book's but I wasn't sure if there may have been other mistakes that I had not noticed. – Valrok Feb 21 '14 at 17:53

3 Answers3

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Indeed you got part (a) right, while the explanation to your solution would be that $A^C$ and $B^C$ are independent.

Note that even though it is quite intuitive that "$A$ and $B$ are independent $\rightarrow$ $A^C$ and $B^C$ are independent", I think it isn't completely trivial.
(You can find some proofs here, though I bet you would succeed in proving it by yourself.)

Oren Milman
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c) (p(A) - p(A n B))/(p(A) + p(B) - p(A n B))

With your numbers

.4 - (.4 * .7) / (.4 + .7 - (.4 * .7))

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(a) This question belongs to "conditional model" But since A and B are independent You may take directly p(B not)=0.3

Reason: independent means

P(A .B)=P(A).P(B)

SIVA NAGA KUMAR.PERUBOYINA
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