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If the probability that Stock A will increase in value [#permalink]

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17 Jan 2012, 11:49

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If the probability that Stock A will increase in value during the next month is 0.54, and the probability that Stock B will increase in value during the next month is 0.68. What is the greatest value for the probability that neither of these two events will occur?

This is a fascinating and tricky question about probability.

This would be a different question if we knew the probabilities were independent --- that is, any increase in Stock A would tell you nothing about whether or not Stock B increased, and vice versa. If the question guaranteed that, then probability P(not A *and* not B) = P(not A)*P(not B) = (1 - 0.54)*(1 - 0.68) = (0.46)*(0.32) = 0.1472 We can't do that here, because the question does not specify that A and B are independent, but I share that because that's also a common type of question on the GMAT.

The best way to think about this particular question is this. Visualize a big circle: that's the space of everything that could happen with these two stocks next year, what statisticians call the "sample space." That represents 100%. The "B space", the area of that space where Stock B increases is 68% of that space, so only 32% is outside the "B space". We don't know the relationship of Stock A & Stock B, but we'd like to know where to put Stock A on the diagram to maximize the area that's outside of both the "A space" and the "B space." Well, with just the "B space" in the big circle, already 68% is taken up, and only 32% is free. If any of the "A space" is outside of the "B space", then it will eat up some of that free 32%. The only way to maximize that free space, to keep all 32% free, would be to put the entirety of "A space" inside "B space", that is, overlapping with "B space." That would be the real world situation in which Stock A rises *only if* Stock B rises: Stock B could go up without Stock A going up, but the the only way that Stock A can go up is if Stock B goes up too. That configuration would leave the maximum amount, 32%, in the region that overlaps with neither "A space" or "B space". Thus, the greatest value for the probability that neither of these two events will occur is 32%, or 0.32. Answer choice = B.

Does that make sense? Please let me know if you have any questions on that.

I think it's easier to understand the concept behind this question with a different example. Say you know that in a certain city on a certain day, the probability that it will be cloudy is 0.4, and the probability that it will rain is 0.2. What is the maximum possible probability that it is both cloudy and that it rains? Well, clouds are a requirement for rain, so every time it rains, presumably it's cloudy. The answer would therefore be 0.2.

The point is that rain and clouds are not independent (unrelated) events - it's not like flipping a coin and then rolling a die. The same can happen with stocks; the changes in the price of one stock might be approximately correlated with the changes in the price of another. Presumably that happens quite often in real life for the stock of two companies in the same industry. So if the probability that stock A does not increase is 0.46, and the probability that stock B does not increase is 0.32, it may be that every time B does not increase, A also does not increase. So the maximum probability that both do not increase is 0.32. Of course, it's also possible that the probability both do not increase is substantially lower than that (it could be as low as 0, in fact).

I'd add that I don't think I've ever seen a real GMAT probability question which tests this idea, so it probably is not important to study in much detail. Fundamentally this question is dealing with overlapping sets (Venn diagrams), but the GMAT questions I've seen don't test overlapping sets using dependent probabilities.
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....So if the probability that stock A does not increase is 0.46, and the probability that stock B does not increase is 0.32, it may be that every time B does not increase, A also does not increase. So the maximum probability that both do not increase is 0.32. Of course, it's also possible that the probability both do not increase is substantially lower than that (it could be as low as 0, in fact).

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.
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....So if the probability that stock A does not increase is 0.46, and the probability that stock B does not increase is 0.32, it may be that every time B does not increase, A also does not increase. So the maximum probability that both do not increase is 0.32. Of course, it's also possible that the probability both do not increase is substantially lower than that (it could be as low as 0, in fact).

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.

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).

Anyway, as Ian mentioned above, this is not a type of question you'll see on the GMAT, so I wouldn't worry about it at all.
_________________

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.

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.

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.

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

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.

[color=#00a651]From what I understand, you mean to say if I had taken .54 in the circle and then applied .68 overlapping to it. I would consume .54 and an additional .12 from the free space. So the probability is still .32[/color]

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...

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.

Re: If the probability that Stock A will increase in value [#permalink]

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11 Nov 2013, 11:23

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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.

....So if the probability that stock A does not increase is 0.46, and the probability that stock B does not increase is 0.32, it may be that every time B does not increase, A also does not increase. So the maximum probability that both do not increase is 0.32. Of course, it's also possible that the probability both do not increase is substantially lower than that (it could be as low as 0, in fact).

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.

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 can not 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).

Anyway, as Ian mentioned above, this is not a type of question you'll see on the GMAT, so I wouldn't worry about it at all.

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.

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

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: http://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
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Re: If the probability that Stock A will increase in value [#permalink]

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31 May 2014, 07:08

Thanks Mike! Thanks for your reply.

I do agree with your analysis.

Only many problem - I someones do notice my ability to solve them intuitively way before I can solve them on paper. Maybe i need to spend more time with counting and probability to develop the same sense.

Re: If the probability that Stock A will increase in value [#permalink]

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28 Dec 2015, 22:11

IanStewart wrote:

I think it's easier to understand the concept behind this question with a different example. Say you know that in a certain city on a certain day, the probability that it will be cloudy is 0.4, and the probability that it will rain is 0.2. What is the maximum possible probability that it is both cloudy and that it rains? Well, clouds are a requirement for rain, so every time it rains, presumably it's cloudy. The answer would therefore be 0.2.

The point is that rain and clouds are not independent (unrelated) events - it's not like flipping a coin and then rolling a die. The same can happen with stocks; the changes in the price of one stock might be approximately correlated with the changes in the price of another. Presumably that happens quite often in real life for the stock of two companies in the same industry. So if the probability that stock A does not increase is 0.46, and the probability that stock B does not increase is 0.32, it may be that every time B does not increase, A also does not increase. So the maximum probability that both do not increase is 0.32. Of course, it's also possible that the probability both do not increase is substantially lower than that (it could be as low as 0, in fact).

I'd add that I don't think I've ever seen a real GMAT probability question which tests this idea, so it probably is not important to study in much detail. Fundamentally this question is dealing with overlapping sets (Venn diagrams), but the GMAT questions I've seen don't test overlapping sets using dependent probabilities.

Probability Problem from OG Practice Test [#permalink]

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29 Feb 2016, 16:06

If probability is 0.54 that Stock A will increase in value during the next month and the probability is 0.68 that Stock B will increase in value during next month. What is greatest possible value for the probability that neither of these two events will occur a) 0.22 b) 0.32 c) 0.37 d) 0.46 e) 0.63

Got stumped by this one. A solution is highly appreciated.

If probability is 0.54 that Stock A will increase in value during the next month and the probability is 0.68 that Stock B will increase in value during next month. What is greatest possible value for the probability that neither of these two events will occur a) 0.22 b) 0.32 c) 0.37 d) 0.46 e) 0.63

Got stumped by this one. A solution is highly appreciated.

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