The bear alarm at Grizzly’s Peak ski resort sounds an average of once

Question

The bear alarm at Grizzly’s Peak ski resort sounds an average of once every thirty days, but the alarm is so sensitively calibrated that it sounds an average of ten false alarms for every undetected bear. Despite this, the alarm only sounds for three out of four bears that actually appear at the resort. On any given day at the resort, what is the approximate probability that there is neither an alarm nor an undetected bear?

(A) 224/365
(B) 13/14
(C) 14/15
(D) 376/390
(E) 29/30

(A) 224/365
(B) 13/14
(C) 14/15
(D) 376/390
(E) 29/30

AaronPond · Accepted Answer

This is one of the more challenging questions you might see dealing with probability. Many people think that this question is confusing, but this confusion is often caused because people don't know how to think about the question. (In fact, think about it for a second: how often do we see GMAT questions that are deliberately worded in a way that confuses those who skim the problem too rapidly?) For example, there are those that believe we have to assume that only one "Bear" or "alarm" event happens per day. This is an unnecessary assumption. Since the problem talks about an "average" rate, we can think about all rates in these terms, without having to worry about the possibility of multiple events happening in a single day. Over time, multiple events in a single day would still be taken into account when we attempt to calculate an "average" rate.

Once you have a strategy that defines how you approach the question, it actually becomes very doable.

Here is the full "GMAT Jujitsu" for this question: We have two mutually exclusive groups (detected vs. undetected bears and alarms vs. no alarms.) If you can organize your thoughts around these mutually-exclusive groups by creating a specialized table, the solution to this problem naturally unfolds. (I call a such tables "Matrix Boxes" in my classes.)

Here is what the blank table might look like:
\[
\begin{matrix}
& \text{Bears} & \text{No Bears} & \text{Totals} \\
\text{Alarms} & \text{____} & \text{____} & \text{____} \\
\text{No Alarms} & \text{____} & \text{____} & \text{____} \\
\text{Totals} & \text{____} & \text{____} & \text{____}
\end{matrix}
\]
Matrix boxes are simplest if we can use concrete numbers. The problem (or the opportunity) here is that we are only given ratios. This means we can invent our own total amount, using leverage inside the problem to guess what number would work the best. The problem states that an alarm sounds "an average of once every thirty days", so our total should be at least a multiple of 30 to avoid messy fractions. But we also want to have it be a multiple of the “total alarms” in order to make the math easy. It looks like the ratios involving total alarms have false alarms in ratios of 10 and real alarms in ratios of 3. So, total alarms should be at least in ratios of 13. Thus, we want a number that is a multiple of \(13\) and a multiple of \(30\). Thus, if we were to pick \(13*30=390\) as our total value, we wouldn't have to deal with ugly math. (Notice, too that \(390\) is the denominator in answer choice D, which might give us a hint here.)

Let's start there:
\[
\begin{matrix}
& \text{Bears} & \text{No Bears} & \text{Totals} \\
\text{Alarms} & \text{____} & \text{____} & \text{____} \\
\text{No Alarms} & \text{____} & \text{____} & \text{____} \\
\text{Totals} & \text{____} & \text{____} & \text{390}
\end{matrix}
\]
The problem tells us that we see 1 alarm happening about every 30 days. Thus over 390 days we get 13 alarms. (We can plug this in to the “Total alarms” slot.) The next two ratios tell us that we get 10 false alarms for every undetected bear (or failed alarm) and that 3 bears are detected for every one bear that is not detected (again, a failed alarm). Notice the “1 undetected bear” is common to both. Plugging all of these values into our Matrix box gives us:
\[
\begin{matrix}
& \text{Bears} & \text{No Bears} & \text{Totals} \\
\text{Alarms} & \text{3} & \text{10} & \text{13} \\
\text{No Alarms} & \text{1} & \text{____} & \text{____} \\
\text{Totals} & \text{____} & \text{____} & \text{390}
\end{matrix}
\]
It is now very easy to fill in the rest of the box:
\[
\begin{matrix}
& \text{Bears} & \text{No Bears} & \text{Totals} \\
\text{Alarms} & \text{3} & \text{10} & \text{13} \\
\text{No Alarms} & \text{1} & \text{376} & \text{377} \\
\text{Totals} & \text{4} & \text{386} & \text{390}
\end{matrix}
\]
The problem wants to know what the “approximate” odds are on any given day that we have neither alarms nor undetected bears. The word “approximate” here is actually a TRAP designed to bait you down a wrong road. Just look down at the answer choices. These are very close together, and trying to approximate the answer would only cause us to miss the question. The problem asks, “what is the…probability that there is neither an alarm nor an undetected bear?” By using our Matrix Box, we can calculate this very quickly:

\(\frac{\text{No Alarms and No Bears}}{\text{Total Days}}=\frac{376}{390}\)

The answer is D.

By the way, for those of you that need extra practice, see these related questions:
https://gmatclub.com/forum/the-bear-ala ... l#p1386648
https://gmatclub.com/forum/the-bear-ala ... l#p2032644
https://gmatclub.com/forum/grizzly-peak ... l#p2033282

IanStewart · Answer

I find the wording of this question very confusing, for quite a few reasons. We need to make some assumptions unmentioned in the question to get the answer (for example, we need to assume that alarms never go off on the same day that a bear sneaks past the alarm system). And I had to read the part about how "sensitively calibrated" the alarm is a few times, because one wouldn't measure that by comparing false alarms to 'undetected bears'. Rather one would compare false alarms to detected bears. And I think it's problematic to ask for an "approximate probability", and then offer answer choices that differ by about 0.0025 (376/390 and 29/30 are only 0.0025 apart). Both answers give the "approximate probability" that the event happens, because both answers are nearly the same value.
.

So I don't think anyone should be concerned if they find the wording confusing. But to solve, you can imagine 4 bears show up. The alarm detects 3 of them, and does not detect 1 of them. We know for every 1 bear the alarm does not detect, the alarm goes off 10 times. So the alarm goes off 3 times when there is a bear, and 10 times when there is not. That is, every 13 times the alarm goes off, there is 1 bear that was not detected.

We know the alarm goes off once every 30 days, so the alarm would go off 13 times in 390 days. We just found out that there will be 1 undetected bear if the alarm goes off 13 times. So in a 390 day period, the alarm will go off 13 times, and there will be 1 undetected bear, so on 14 days there will be an alarm or an undetected bear, and thus on 376 days there will be neither. So the answer is 376/390.

tjerkrintjema · Answer

Wow I had to read the question so many times, and i still get lost.
Would this question still classify under the 600-700 range or maybe because of the confusing language fall in the 700 range?

Erjan_S · Answer

I did find this question to be both confusing and clumsy, but I could not articulate reasons for this opinion as clearly, as you did! It is not a real GMAT question I think, do they do better job formulating actual GMAT questions?

AnthonyRitz · Answer

Erjan et al,

This question is not well-worded and is, I believe, retired by Veritas for that reason. The occasional clunker question happens to the best of us; sorry! You can expect to face questions on the actual GMAT that are clearer. I wouldn't worry about this one or its siblings (it comes from a set of four related to the same fact pattern).

ScottTargetTestPrep · Answer

Let x denote the number of undetected bears in the facility.

Since there are 10 false alarms for every undetected bear, the number of false alarms is 10x.

We are given that the alarm sounds for three out of four bears that actually appear at the resort, which means one out four bears is undetected. Since the number of undetected bears is x, the total number of bears is 4x and the number of detected bears (i.e. the number of times where the alarm sounds and there is indeed a bear in the facility) is 4x - x = 3x.

So, the alarm sounds 10x + 3x = 13x times (10x of which is false). Since the alarm sounds an average of once every thirty days, we can approximate the number of days until the alarm sounds 13x times by (13x)(30) = 390x.

Out of the 390x days, there are only 13x days with an alarm sounding and x days with an undetected bear. Thus, for 390x - (13x + x) = 376x days, there is neither an alarm sounding nor an undetected bear. Therefore, the approximate probability that there is neither an alarm sounding nor an undetected bear is 376x/390x = 376/390

Answer: D

UncleFrodo · Answer

Hi Aaron,

Thanks for sharing your way of solving question! It helps me to crack the question. However, I was a bit confused with this:
 
and this

I didn't understand why we should  invent total amount . Also it wasn't clear why it should be  13*30=390 as our total value  and why you called it "total value". Or what "total value" of what?
I mean that we've got all info from the task and there is no ways of inventing something. And if we invent then what it symbolizes?
It is crystal clear that your matrix and math correct, but for me was a bit illogical. 
But then I discover why i was so confused.
If we eliminate the idea of "invent our own total amount" (I am sorry for that) and say that we need to figure out number of events based on shared info than it works for me.

For instance, we know that once per 30 days we got an alarm.  Alarm is specific event . On a flip side we got 29 our 30 days without alarm, so lets call it  silence event . 
We got 1 alarm event per 30 days, hence 13 alarms, which is the total number of alarms, we will get per 390 days.

To be more clear for this task we must say that each of this events (alarm of silence) can be true or false. 
We know that there:
1. 3 true alarms (the alarm only sounds for three out of four bears that actually appear at the resort)
2. 10 false alarms (an average of ten false alarms for every undetected bear)
3. 1 false silence (the alarm only sounds for three out of four bears that actually appear at the resort)
4. X true silence (there is neither an alarm nor an undetected bear)
out of 390 events.
X = "total events" - "true alarms" - "false alarms" - "false silence"
X = 390 - 3 - 10 - 1 = 376 
The answer is 376 true silence events out of 390 days -> 376/390

UncleFrodo · Answer

Hi Scott, I think you posted a correct solution but for a different question.

ScottTargetTestPrep · Answer

Good call! Just adjusted my solution. Thanks!

AaronPond · Answer

Great question, Uncle Frodo! The approach that I shared in my initial solution is actually a phenomenal strategy for a    whole class    of questions on the GMAT. I call this strategy "  Easy Numbers  " in my classes. Often, questions that give you ratios without total amounts feel very abstract. By definition, a probability is:

\frac{	ext{# of occurrences of a specific event }}{	ext{Total number of possible occurrences}}

Thus, if we want to calculate a probability, we need to know the total number of occurrences. However, in terms of this specific question, calculating a probability for "any given day" is a little nebulous. We have an ongoing process, with various, overlapping rates ("once every thirty days"... "ten false alarms for every undetected bear", etc.). We need to calculate an average daily rate, but what "total" will we use to actually calculate the probability? Total in a month? Total in a year? By choosing a number that makes the math easy, we turn a very abstract concept into a concrete, solvable approach. There won't be an integer count of "  no alarms and no bears  " in only 30 days. But in  390  days, we  will  have an integer count. As you can see,  390  is not an arbitrary number. It is the the number of days that allows us to get a pretty, "no-weird-decimal value" for the "  no alarms and no bears  " count that we need in the top part of the probability fraction. This is why chose  390  as my "total number of possible occurrences" value. It made for easy math.

As you can see, though, I was also trying to help as many readers as possible to not only answer this question, but see the strategic approach to many similar questions. "  Easy Numbers  " is powerful GMAT Jujitsu. It cuts through the crap of abstraction and gives test takers concrete leverage to solve    any    question that deals with percents or ratios without total amounts.

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The bear alarm at Grizzly’s Peak ski resort sounds an average of once

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