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In the table below, choose the two numbers - Two-Part Analys [#permalink]

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20 Sep 2012, 05:14

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In the table below, choose the two numbers that are consistent with the information that is given. In the first column, select the row that shows the probability that at least one of the events A and B occurs, and in the second column, select the row that shows the probability that event B occurs.

Please explain how to approach such question.

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In the table below, choose the two numbers that are consistent with the information that is given. In the first column, select the row that shows the probability that at least one of the events A and B occurs, and in the second column, select the row that shows the probability that event B occurs.

Please explain how to approach such question.

Conty911,

I'm happy to help, but I believe something is funky about this question. Are you sure there wasn't any other text preceding the problem? There just doesn't seem to be enough information to give a sensible answer.

I am not particularly impressed with the quality of the question either --- the complicated phrase "at least one of the events A or B occurs" is a convoluted way of saying "A or B" --- in logic (at least according to the conventions GMAT follows) the word "or" implies one or the other or both --- saying "A or B happens", means A happens, or B, or both. You don't have to specify the "at least" part at all --- it's totally redundant.

Now, it's undeniably true that

P(A or B) > P(B)

so the entry in the left column must be lower than the entry in the right. At the moment, though, in the absence of any other conditions, it's not particularly clear to me that the answer couldn't be any pair in which the entry in the the left column were lower than the entry in the right. After all, P(A) could be anything, and the overlap region, P(A and B), could be anything. The requirement of P(A or B) > P(B) seems to be the only constraint in this problem. That would produce fifteen distinct right answer combinations.

Something must be missing. What is the source? What is the exact text of the question? I feel like there is some crucial condition missing.

Please let me know if you can uncover any further parts to this question.

Mike
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Mike McGarry Magoosh Test Prep

Education is not the filling of a pail, but the lighting of a fire. — William Butler Yeats (1865 – 1939)

Re: In the table below, choose the two numbers - Two-Part Analys [#permalink]

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09 Oct 2012, 12:05

mikemcgarry wrote:

conty911 wrote:

In the table below, choose the two numbers that are consistent with the information that is given. In the first column, select the row that shows the probability that at least one of the events A and B occurs, and in the second column, select the row that shows the probability that event B occurs.

Please explain how to approach such question.

Conty911,

I'm happy to help, but I believe something is funky about this question. Are you sure there wasn't any other text preceding the problem? There just doesn't seem to be enough information to give a sensible answer.

I am not particularly impressed with the quality of the question either --- the complicated phrase "at least one of the events A or B occurs" is a convoluted way of saying "A or B" --- in logic (at least according to the conventions GMAT follows) the word "or" implies one or the other or both --- saying "A or B happens", means A happens, or B, or both. You don't have to specify the "at least" part at all --- it's totally redundant.

Now, it's undeniably true that

P(A or B) > P(B)

so the entry in the left column must be lower than the entry in the right. At the moment, though, in the absence of any other conditions, it's not particularly clear to me that the answer couldn't be any pair in which the entry in the the left column were lower than the entry in the right. After all, P(A) could be anything, and the overlap region, P(A and B), could be anything. The requirement of P(A or B) > P(B) seems to be the only constraint in this problem. That would produce fifteen distinct right answer combinations.

Something must be missing. What is the source? What is the exact text of the question? I feel like there is some crucial condition missing.

Please let me know if you can uncover any further parts to this question.

Mike

I assume P (a+b) = 1/2 of P(b) so 0.5 and 0.25 is the answer. But not sure if my assumption is true.

I assume P (a+b) = 1/2 of P(b) so 0.5 and 0.25 is the answer. But not sure if my assumption is true.

Dear Ravipprasad,

The trouble is --- that's one possible answer, but not the only one.

It could be true that P(A) = 0.25 and P(B) = 0.25 and P(A or B) = 0.50 or that P(A) = 0.25 and P(B) = 0.10 and P(A or B) = 0.35 or that P(A) = 0.10 and P(B) = 0.25 and P(A or B) = 0.35 or that P(A) = 0.40 and P(B) = 0.10 and P(A or B) = 0.50 etc. etc.

We have no guarantee that P(A) = P(B). Furthermore, I was assuming in these four cases that A & B are disjoint, that is to say, that they have no overlap. If A & B are disjoint, then P(A or B) = P(A) + P(B) If A & B are not disjoint, then P(A or B) = P(A) + P(B) - P(A and B)

For example, it could be true that: P(A) = 0.25, P(B) = 0.50, P(A and B) = 0.15, so P(A or B) = 0.60

It actually would be an excellent GMAT math question simply to count all the possible answers one could make on this chart! Either the person who wrote the question was not good at writing questions, or the person who posted it omitted crucial information. Be careful making assumptions --- in a real, well-written GMAT problem, no assumptions will be necessary, and in fact, making assumptions will just get you in trouble.

Does all this make sense?

Mike
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Mike McGarry Magoosh Test Prep

Education is not the filling of a pail, but the lighting of a fire. — William Butler Yeats (1865 – 1939)

Re: In the table below, choose the two numbers - Two-Part Analys [#permalink]

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11 Jun 2013, 09:08

The question posted here is incomplete. The question (from source) clearly mentions "The events A and B are independent, and the probability that event A occurs is 0.4." and then the rest of the question.

The question posted here is incomplete. The question (from source) clearly mentions "The events A and B are independent, and the probability that event A occurs is 0.4." and then the rest of the question.

Dear GmatYes, Thank you!! That makes a ton of sense! I was wondering what kind of source would present such a lame half-baked question. With this new information, it's actually a sensible question at last! Thank you very much! Mike
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Mike McGarry Magoosh Test Prep

Education is not the filling of a pail, but the lighting of a fire. — William Butler Yeats (1865 – 1939)

We're asked to find the probability that event B occurs, such that one of the answer choices reflects the probability of either A or B occurring—in other words, any outcome other than neither A nor B occurring.

Since the only outcome we want to exclude is neither A nor B occurring (see stimulus explanation, above), we can create an equation by subtracting that one undesired outcome from the total of 1:

Probability that at least one of A and B occurs = 1 – [0.6(1 – b)]

= 1 – (0.6 – 0.6b)

= 1 – 0.6 + 0.6b

= 0.4 + 0.6b

(Note that there are two other ways to set this equation up. We could add the three desired outcomes together to yield 0.4b + 0.4(1 – b) + 0.6b. Alternately, we could add the probabilities of A and B, then subtract the double-counted outcome of both occurring, yielding 0.4 + b – 0.4b. On Test Day, go with whichever approach feels most straightforward—all of these approaches simplify to 0.4 + 0.6b.)

Now let's consider the possible values given in the answer choice list. Thinking about the situation logically, we can determine that the probability that B occurs must be less than the probability that at least one of A or B occurs. We will therefore begin by plugging in the smaller values from our list for the variable b in the equation we set up above. If one of these values for b results in a probability for "at least one of the events A or B" that matches one of the given choices, we know we have found the correct answer.

If b were equal to 0.1, then the probability that at least one of A and B occurs would equal 0.46, which is not an answer choice. This strongly suggests that 0.25 is the probability that B occurs, since it is the only other small answer choice. A larger probability might be possible, since the probability of at least one of A and B might be as high as 0.8, but plugging in b = 0.25 is a reasonable next step.

We'll substitute b = 0.25 into our equation and see whether it produces one of the other answer choices. If it does, that pair will be correct. If it doesn't, we'll try a different value for b. Plugging b = 0.25 into our equation yields:

Probability that at least one of A and B occurs = 0.4 + 0.6(0.25)

= 0.4 + 0.15

= 0.55

This possible value matches one of the answer choices, so we have found the values that satisfy the conditions. Now, as always with Two-Part Analysis questions, we must be careful to select the correct answer choices in the appropriate columns; it would be a shame to do all the math correctly but lose the points for this question by mixing up the columns.

The correct response for the "probability that at least one of the events A and B occurs" is 0.55; the correct response for the "probability that event B occurs" is 0.25.