Lesson: General GMAT Probability Strategies

Comment on General GMAT Probability Strategies

Hi Brent, Can you help me understand the following 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.

1. If the alarm sounds, what is the probability that a bear has actually been sighted?
2. In any given day at the resort, what is the approx. probability that there is neither an alarm nor an undetected bear?
3. If the alarm were to sound an average of ten false alarms for every detected bear, the probability that a sounded alarm would indicate an actual bear would be?
4. Approx. how many bears appear in the resort each year?
gmat-admin's picture

NOTE: I'm not crazy about the wording of any of these question, but here goes....

QUESTION 1

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.

1. If the alarm sounds, what is the probability that a bear has actually been sighted?

The alarm only sounds for three out of four bears that actually appear at the resort.

So, for every 4 bears, 3 are detected by the alarm system and 1 is not.

Total number of alarms for the 3 detected bears = 3.

There are 10 false alarms for every undetected bear. So, the number of alarms from the 1 undetected bear = 10.

Total number of alarms = 13

So, for every 13 alarms, 3 are real and 10 are false.

P(an alarm = an actual bear) = 3/13

gmat-admin's picture

QUESTION 2

This isn't a very good question. I'd skip it.
More here: https://gmatclub.com/forum/the-bear-alarm-at-grizzly-s-peak-ski-resort-s...

Where did we get the 14 here from?


"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."
gmat-admin's picture

I don't think there's much benefit in examining question #2 any further. It's not a great representation of what you'll see on test day (which is probably why Veritas pulled the question from circulation).

gmat-admin's picture

QUESTION 3
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.

3. If the alarm were to sound an average of ten false alarms for every detected bear, the probability that a sounded alarm would indicate an actual bear would be?

Go back to the information from question 1

For every 4 bears, 3 are detected by the alarm system and 1 is not.

Total number of alarms for the 3 detected bears = 3.

There are 10 false alarms for every DETECTED bear. So, the number of alarms from the 3 undetected bears = 30.

Total number of alarms = 33

So, for every 33 alarms, 3 are real and 30 are false.

P(an alarm = an actual bear) = 3/33 = 1/11

But the question says that for every UNDETECTED bear there are 10 false alarms.
gmat-admin's picture

Yes, that's what the original question states. However, question 3 asks: "If the alarm were to sound an average of ten false alarms for every DETECTED bear, the probability that a sounded alarm would indicate an actual bear would be?"

gmat-admin's picture

QUESTION 4

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.

4. Approx. how many bears appear in the resort each year?

From question #1, we see that, for every 13 alarms, 3 are real and 10 are false.

The question tells us that there is 1 alarm every 30 days.

So, we'll get 13 alarms in 390 days.

In those 390 days (with 13 alarms), there are 3 detected bears and 1 undetected bear.

So, in 390 days, a total of 4 bears visit the resort.

So, in 365 days, there will be a slightly fewer than 4 bear visits.

Hi Brent-can you give example where outcomes are not equally likely?

https://gmatclub.com/forum/if-two-points-a-and-b-are-randomly-placed-on-the-circumference-234196.html

Confused. My approach -
Chord = line thru the circle and longest chord = diameter.

Diameter = 12 hence chords can be 1 - 12.
So essentially the problem is - what is the probability to get at least 6 from a set of numbers from 1 - 12.
Which should be
P(6) ... P(12) = 7/12
Where did I go wrong ?
Please let me know

gmat-admin's picture

I love this approach!! The only problem is that each possible chord length (from 0 to 12) is not equally likely.

For example, if we randomly choose a second point, we are twice as likely to get a chord that is greater than 6 than less than 6.

So, even though the chord length can range from 0 to 12, we cannot assume that each outcome (chord length) is equally likely.

Here's an analogous case: Harry, a teenager from Canada, wants to determine the probability that he will marry German Chancellor Angela Merkel some day.

There are two possible outcomes:
1) Harry will marry Angela Merkel some day
2) Harry will NOT marry Angela Merkel some day

Since there are 2 possible outcomes, P(Harry marries Angela Merkel) = 1/2

As we might expect, the two outcomes are not equally likely. So, we can't use this approach.

Thanks that helps !

Hi Brent,

Please refer the Q : https://gmatclub.com/forum/jill-has-applied-for-a-job-with-each-of-two-different-companies-what-220335.html

Could you please explain a detailed solution using the techniques taught by you? I am getting really confused with "AT LEAST, "AT MOST" and "EXACTLY"

Thanks & Regards,
Abhirup
gmat-admin's picture

Phew, that one took a while!
Here's my full solution: https://gmatclub.com/forum/jill-has-applied-for-a-job-with-each-of-two-d...

Cheers,
Brent

Thanks Brent for that wonderful insight.
I assumed P(Getting job from A & Getting Job from B) = 1-P(Not Getting job from A & not getting job from B)

Isn't P(Doing a task X) = 1 - P(Not doing a task X)? Please help me understand the flaw in my reasoning.
gmat-admin's picture

You're correct to say that P(Doing a task X) = 1 - P(Not doing a task X)

However, the task in this case involves TWO events.

Let's say that a certain club will only admit people who are left-handed AND own a motorcycle.

So, if Joe is admitted to the club, we know that Joe is left-handed AND he owns a motorcycle.

What if Joe is NOT admitted to the club?
What can we conclude?
We know that Joe did not meet the admission criteria (which requires left-handedness AND motorcycle ownership)

So, ONE possibility is that Joe is RIGHT-handed, and he does NOT own a motorcycle.
However, there are 2 more possibilities:
- Joe is RIGHT-handed AND he owns a motorcycle.
- Joe is left-handed AND he does NOT own a motorcycle.

The same applies to the question we're talking about.

Does that help?

Cheers,
Brent

So that means to get P(Job offers from both companies) we have to eliminate all the probabilities that have either company offering job or neither offering job i.e. 1 -[ P(job from A but not B) +P(job from B but not A) + P(no job from both) ]. This is how we achieve P(Not doing the task) in this case.
gmat-admin's picture

That's correct.

https://gmatclub.com/forum/if-two-points-a-and-b-are-randomly-placed-on-the-circumference-234196.html

I do not understand this solution. Why are we taking ratio of angles.
gmat-admin's picture

Question link: https://gmatclub.com/forum/if-two-points-a-and-b-are-randomly-placed-on-...

Great question!

KEY CONCEPT: the length of a circle's arc is proportional to its central angle.
For example, if we have a circle with radius 10, then an arc with a central angle of 180° (i.e., half the circle) will be TWICE the length of an arc with a central angle of 90° (i.e., one-quarter) of the circle.

So, rather than compare the lengths of the desired arcs, we can just compare the central angles.
That said, if we were to calculate the arc lengths instead, we'd arrive at the same probability.

Does that help?

Cheers,
Brent

Hi Brent, would you mind explaining again the difference between mutually and not mutually exclusive events please? Thanks very much x
gmat-admin's picture

Those two concepts are covered here: https://www.gmatprepnow.com/module/gmat-probability/video/746

Once you review those videos, please let me know whether you need an clarification.

Cheers,
Brent

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