Lesson: General GMAT Probability Strategies

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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 !

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