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Is the probability P(A and B) > 1/3?

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by Mike@Magoosh » Tue Sep 24, 2013 1:47 pm
Suppose A and B are two events. Is the probability P(A and B) > 1/3?

Statement #1: P(A) = 0.8 and P(B) = 0.7

Statement #2: P(A or B) = 0.9

For a discussion of DS problems about Probability, as well as the OA & solution to this question, see:
https://magoosh.com/gmat/2013/gmat-data- ... obability/

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Source: — Data Sufficiency |
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by theCodeToGMAT » Wed Sep 25, 2013 12:41 am
Statement 1: The Event A & B are independent events.. if they were dependent that p(a) + p(b) should be less than or equal to 1..

So. for independent events. P(A and B) = P(A) x P(B) = 0.8 * 0.7 = .56..
SUFFICIENT

Statement 2:
p(A or B) = 0.9
Events can be dependent or independent

Suppose the event is Independent: P(A or B) = P(A) + P(B) - P(A and B)

--> P(A and B) = P(A) + P(B) - P(A or B)

Let p(a) be 0.9 and p(b) be 0.9

So, P(A and B) = 0.9 .. YES

Let p(a) be 0.4 and p(b) be 0.6

So, P(A and B) = 0.1 .. NO
INSUFFICIENT

Answer [spoiler]{A}[/spoiler]
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by Mike@Magoosh » Wed Sep 25, 2013 1:37 pm
theCodeToGMAT wrote:Statement 1: The Event A & B are independent events.. if they were dependent that p(a) + p(b) should be less than or equal to 1..
Dear Rahul,
With all due respect, what you say there is not correct. The fact that P(A) + P(B) > 1 tells us that the two events can't be mutually exclusive, which would play out in the OR rule, but we can deduce nothing about independence.

Consider --- suppose
A = on a given day, there is no rain in Khartoum
B = on a given day, the net change in Dow Jones index is less than $2000

Both A and B have probabilities well over 0.9, so their sum is clearly well above 1, but those two events have absolutely no influence on one another, so they are still completely independent, despite having a sum of probabilities well above one.

Does this make sense?
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by theCodeToGMAT » Wed Sep 25, 2013 8:14 pm
Hello Mike,

Thanks for this useful information.

I solved this question considering the fact stated by you:

".... but those two events have absolutely no influence on one another, so they are still completely independent...."

Also, if the events are dependent then the probability is always one, as probability takes into consideration the total number of possible outcomes and total number of ways..... please correct me if i am wrong
For example:
A -> Probability of heads = 1/2
B -> Probability of tails = 1/2
and the sum is 1.

Please correct me if I am wrong....
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