If the probability that Stock A will increase in value during the next

Responding to a pm.

The probability that stock A does not increase is 0.46, and the probability that stock B does not increase is 0.32. Now, how can the probability that both do not increase be more than individual probability of not increasing for each? So the probability that both do not increase cannot be more than 0.32. Basically the probability that both do not increase is between 0 and 0.32, inclusive (in fact the moment you realize this, you have the correct answer right away).

Please check the attachment

Recall that if set U is the universal set containing two sets A and B, we have:

P(U) = P(A) + P(B) - P(A and B) + P(neither A nor B)

Since P(U) = 1, we can also say:

1 = P(A) + P(B) - P(A and B) + P(neither A nor B)

If we let the probability that stock A will increase be P(A) and the probability that stock B will increase be P(B), then we are given that P(A) = 0.54 and P(B) = 0.68. Thus, we can say:

1 = 0.54 + 0.68 - P(A and B) + P(neither A nor B)

Notice that we are being asked for the greatest value of P(neither A nor B). If that is the case, we want P(A and B) to be as large as possible since we are subtracting it. However, notice that P(A and B) can’t be larger than either P(A) or P(B). Therefore, P(A and B) can be only as large as the lesser value of P(A) and P(B). So, here P(A and B) can be as large as P(A), or 0.54. Thus:

1 = 0.54 + 0.68 - 0.54 + P(neither A nor B)

1 = 0.68 + P(neither A nor B)

0.32 = P(neither A nor B)

Answer: B

Dear b2bt,
I'm happy to respond.

As I am sure you appreciate, there is often more than one valid way to approach a problem. For this particular problem, in the section I marked in green above, you demonstrated a perfectly correct way to think about this starting from the probabilities of each event happening. That is a perfectly correct approach, and it results in a correct answer.

In some of the entries above, I and others started instead with the probability of each event NOT happening. This can also lead to a correct answer. Let's say
A = the event that stock A increases next month
B = the event that stock B increases next month
The prompt tell us
P(A) = 0.54
P(B) = 0.68
From here, we can calculate
P(not A) = 1 - 0.54 = 0.46
P(not B) = 1 - 0.68 = 0.32
We want the maximum overlap of P(not A) and P(not B), so even if they have full overlap, the size of that overlap region could only be as big as P(not B) = 0.32. This approach also logical leads us to the answer.

It's important to appreciate that there are two different paths of logic that lead to the answer. In fact, I would: if you only have one way of thinking about any GMAT math problem, you don't really understand it. Multiple approaches in problem solving are so important for deep understanding of mathematics.

Does all this make sense?
Mike

Dear Rohan,
Here's a series of blogs you may find helpful:
https://magoosh.com/gmat/2012/gmat-math- ... ity-rules/
https://magoosh.com/gmat/2012/gmat-math- ... -question/
https://magoosh.com/gmat/2013/gmat-proba ... echniques/
I hope all this helps.
Mike

Wow! I cannot believe I gotta break out my prob books and revise conditional prob for this . Kudos and here is a link:

https://en.wikipedia.org/wiki/Conditional_probability

The crucial point here is correlation, not causality.

Rains and clouds are correlated because rains cause clouds. Not all things that are correlated have a causal relationship.

Sales of suntan lotion and sales of ice cream are correlated --- not because one causes the others (folks using suntan lotion as an ice cream topping??), but rather both are caused by hot sunny weather.

Much in the same way, Stock A's increase doesn't cause Stock B's increase, nor vice versa, but rather all the market conditions in general (interest rates, price of oil, whether Greece will go bankrupt, etc. etc. etc.) --- everything that moves the market will cause all stocks to up together or all stocks to go down together. In fact, if you follow the stock market, you know that's very much how stocks move.

That's why it may be that every time B increases, A also increases. Not that one causes the other, but both are driven by a mutual underlying cause. That's another way that two things can be correlated.

Does this make sense? Please let me know if you have any further questions.

Mike

Hi,
there are a lot of solutions talked of above..
But the crux of the entire thing is...
If a Q is asking for the greatest value possible, we can infer there can be various possible values..

GREATEST:- when will it be greatest, when there is complete overlap..
that is P of both happening = P of one of them happening(ofcourse it will be lower)
so, teh P of both not happening is 1-the larger prob= 1-.68=0.32

LOWEST:- If, say, we are to find the lowest possiblity, when there is no/least possible overlap..
so P of both happening= .68 + .54 -1 = .22..
so least prob of both not happening= 1-(.68+.54-.22)=0..

I have one ultimate question in my mind!
How is that all (or almost all) the official GMAT Prep questions in the latest Exam Pack 1 released by GMAC on September 2013 has already been discussed on GC/BTG at a much earlier date i.e., some questions almost dating back to 7-8 years. Since, GMAC claims that there is no overlap with any of the old Questions it has released so far, from where do these questions surface?

In the very discussion thread above, I can see that the experts rejecting the possibility of seeing such a question in the official test, I encountered the same Q in GMAT Prep Exam Pack 1 - Test #4. It was my 12th question, after getting the first 11 questions correct.

Any light on this would be much helpful!

1.Assume a period of 100 days. So 54 days stock A will increase and 68 days stock B will increase
2. The 54 days of increase in stock A can be contained within the days of increase in stock B.
3. So the maximum number of cases when both the events do not happen is the remaining after subtracting the days of B's increase only which is 100-68=32 days
4. So the greatest probability is 32/100=0.32

Hi Bunuel

This is how I figured this problem. Can you please correct me if I am wrong.

We don't know here that Events A and B are independent. So Probability that A occurs will be
P(A) = P(only A occurs) + P(A and B both occur) = 0.54
Similarly,
P(B) = P(only B occurs) + P(A and B both occur) = 0.68

Now this becomes simply a problem of SETS and we have to minimize the ONLY A and ONLY B part of the vein diagram while maximizing the Common area.
Clearly, P(only A) should be 0 and P(only B) should be 0.14 and P(A and B both occur) should be 0.54...This can be easily done by drawing a vein diagram.

Using Venn Diagrams is also a great way to solve such a question.

In relation to the explanation provided by IanStewart, I understand the example about the rain and clouds because there is a causality relationship behind (clouds are necessary in a rain). But in the original question, we don't know whether A depends on B or B depends on A. In this sense, we could also say that every time A does not increase, B also does not increase (the opposite stated by Ian). Consequentely, we could say that the maximum probability that both do not increase is 0.46 (which is also a choice (D)).

How could we figure out that the correct causality relationship is the mentioned by Ian?. Please explain.

Dear Vin,
I'm happy to respond.

This is a very funny thing about probability and counting problems, more than many other branches of math. How to explain this? Think about, say, algebra. In algebra, there are very fixed rules, and for such-and-such a problem, there's not much ambiguity about what steps will lead to the answer --- yes, for folks still learning, there might be confusion, but for anyone familiar with the algebra, there's really no debate about what to do: there's a clear linear path to the answer. Algebra depends heavily on left-brain skills: logic, organization, procedures, and precision. If you know the rules & formulas of algebra, you know well over 90% of what you need to know to solve problems.

Probability and counting are two branches of math that are not like this at all. Yes, there are some rules & formulas, but even when you completely know those rules & formulas, you really know only about 20% of what you need to know. What's more important in these branches is perspective and ability to frame the problem in your mind. Instead of the left-brain focus of "what to do?" we have to start with the right-brain focus of "how to see?" --- when a probability or counting problem is properly framed, properly viewed, then what to do becomes quite straightforward.

You ask for resources. This is VERY tricky. You see, with left-brain skills, rules & procedures, we can just give explicit steps --- "do this, then do this, then do that." Left-brain skills lend themselves well to recipes, methods, and step-by-step instructions. Right-brain skills of intuition and pattern-matching are not like that at all --- there's often no quick way to summarize it. You have to develop it though experience, over time. See this blog for more on left-brain/right-brain skills in math:
https://magoosh.com/gmat/2013/how-to-do- ... th-faster/

Magoosh actually provides a HUGE resource here. It's true, we have a few probability video lessons that cover the basic rules, but there's no way those lessons could cover all the right-brain skills needed for seeing problems the way you need to see them. The BIG resources are the video explanations following each and every practice question. That's really where a student has the opportunity to see --- after doing a problem for himself, the student can witness how the instructor frames the problem. Students often miss this --- they are so focused on "what to do" that they don't give sufficient attention to the perceptual choices with which the instructor begins the analysis of the problem. Developing a deep understanding of probability involves thinking critically about these perceptual choices --- why did the instructor look at it this way rather than that way? Those are great questions to ask if you can't figure it out on your own. The fact that every Magoosh question has its own VE is tremendously valuable resource that, unfortunately, some less perceptive students completely underestimate.

For probability questions you found outside Magoosh, say here on GMAT club, also pay attention to these perceptual choices that experts make, the very first step they take in framing the problem. Don't focus simply on "what did they do" --- focus first and foremost on "how did they look at the problem? how did they frame it?" Again, if you are unclear, ask. You are always more than welcome to solicit my input on any problem: just send me a private message with a link.

I know this is probably a less satisfying answer than you were hoping to receive. Does all of this make sense?
Mike

From what I understand, you mean to say if I had taken .54 in the circle and then apllied .68 overlapping to it. I would consume .54 and an additional .12 from the free space. So the probabilty is still .32

I still dont understand cause we are talking about probability of "not increasing"
If A doesn't increase the not increasing probability is .46 and as it is not increasing B will also not increasing so it will not consume any area.

Really sorry to confuse so much...

Wow - now that this is a GMAT Prep question - Must look at these concepts in more detail..

Hi Mike - Thanks for your clarifications!

None of the Magoosh tutorials cover this problem. I just encountered this in my GMAT Prep Exam - 4.

Can you please point me to some resource on where i can learn more about these concepts?

Vin