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# Among a group of 2,500 people, 35 percent invest in municipal bonds, 18 percent invest in oil stocks, and 7 percent invest in both municipal bonds and oil stocks. If 1 person is to be randomly selected from the 2,500 people, what is the probability that the person selected will be one who invests in municipal bonds but NOT in oil stocks? GMAT Explanation

Among a group of 2,500 people, 35 percent invest in municipal bonds, 18 percent invest in oil stocks, and 7 percent invest in both municipal bonds and oil stocks. If 1 person is to be randomly selected from the 2,500 people, what is the probability that the person selected will be one who invests in municipal bonds but NOT in oil stocks?

(A) 9/50
(B) 7/25
(C) 7/20
(D) 21/50
(E) 27/50

Combinatorics questions, especially ones tangled up with probability, scare students. Can there be a very challenging probability question? Yes! Are most of them tough? Not really. Do you need a bunch of complex math to solve GMAT probability questions? Nope. Let's see what we need to get this one solved.

## Define the question

What is the probability that the person selected will be one who invests in municipal bonds but NOT in oil stocks?

OK, not so bad. Munis but no oil stocks. So what is probability exactly? Specific Scenario/Total Scenarios. So Muni/Not Oil/Total. This is starting to feel like overlapping sets with a side of probability...

## Let's get the information organized!

Let's call the 2500 people 100 people. It won't make a difference in the answer because everything is based on percents/proportions and will make it easier to do the calculations. So:

Municipal Bonds: 35

Oil Stocks: 18

Muni and Oil: 7

Looks like we have a two group overlapping set. We're going to need a diagram.

So just fill in the given information and then make easy inferences. You'll be able to get the whole chart filled in. You end up with 28 people that have municipal bonds but not oil stocks. But, 28 isn't the answer because we're asked about probability. So take 28 and put it over the total. What's the total? The 2500. BUT we changed that to 100. 28/100 = 7/25.

## More GMAT two group overlapping sets and probability practice!

Here's a challenging sets question from the GMAT Prep Tests 1 and 2. Stay organized. Make the easy inferences and you should be fine.

In a certain city, 80 percent of the households have cable television, and 60 percent of the households have videocassette recorders. If there are 150,000 households in the city, then the number of households that have both cable television and videocassette recorders could be any number from:

Here's one from GMAT Question of the Day that's very similar in that the answer is based on proportions. It doesn't have a probability component but again: same idea and great practice.

GMAT Question of the Day 2 group overlapping sets proportions and percents

This one is a bit different but will still give you 2 group practice and will give you exposure to overlapping sets on data sufficiency which often comes down to counting equations.

GMAT Question of the Day 2 group overlapping sets Data Sufficiency

And here's a very tough DS overlapping sets questions. Again, deals with counting equations/system of questions.

GMAT Question of the Day Challenging 2 Group Overlapping Sets Data Sufficiency Counting Equations

# In a certain city, 80 percent of the households have cable television, and 60 percent of the households have videocassette recorders. If there are 150,000 households in the city, then the number of households that have both cable television and videocassette recorders could be any number from: GMAT Explanation + Additional Practice Questions!

In a certain city, 80 percent of the households have cable television, and 60 percent of the households have videocassette recorders. If there are 150,000 households in the city, then the number of households that have both cable television and videocassette recorders could be any number from:

(A) 30,000 to 90,000 inclusive
(B) 30,000 to 120,000 inclusive
(C) 60,000 to 90,000 inclusive
(D) 60,000 to 120,000 inclusive
(E) 90,000 to 120,000 inclusive

Video explanation below! This is a 2 group overlapping sets question. How do I know? Well, you've got two categories, Cable Television and Videocassette Recorders. Households are either in cable, video, both, or neither. This sorting of people or things into two categories usually indicates a two group overlapping sets question. These are best solved using a 2 group matrix (see diagram below).

## Let's get organized!

The first step is to place your actual numbers into the chart. The only one we start with is 150,000. You can safely chop off the zeros to compact a bit. Take 80% and 60% of 150 to derive total Cable Television and total Videocassette Recorders and make as many inferences as you can to fill in the chart. With numbers in place you can see that the Videocassette and Cable TV box can't be more than 90 people because total Videocassette is 90 (if the total is 90 one of the components of the total can't be more than 90). So the max households that have both Cable Television and Videocassette Recorders is 90. Since total Cable TV is 120, and the max Cable TV but not Videocassette is 60 the min both Cable TV and Videocassette could be is 60. So the number of households that have both cable television and videocassette recorders could be any number from 60,0000 to 90,0000. C.

## More GMAT overlapping sets practice questions!

Here's an overlapping sets question that adds in a bit of probability. Don't let the probability component get in your way! Often, probability questions are easy. Take it step by step.

Among a group of 2,500 people, 35 percent invest in municipal bonds, 18 percent invest in oil stocks, and 7 percent invest in both municipal bonds and oil stocks. If 1 person is to be randomly selected from the 2,500 people, what is the probability that the person selected will be one who invests in municipal bonds but NOT in oil stocks?

Here's one from GMAT Question of the Day that's in the same vein. It also has percents/proportions and will give excellent practice using the two group overlapping sets matrix.

GMAT Question of the Day 2 group overlapping sets proportions and percents

This one is a bit different but will still give you 2 group practice and will give you exposure to overlapping sets on data sufficiency which often comes down to systems of questions/counting equations.

GMAT Question of the Day 2 group overlapping sets Data Sufficiency

And another challenging Data Sufficiency overlapping sets question. Again, deals with counting equations/system of questions.

GMAT Question of the Day Challenging 2 Group Overlapping Sets Data Sufficiency Counting Equations

A survey was conducted to determine the popularity of 3 foods among students. The data collected from 75 students are summarized as below

48 like Pizza
45 like Hoagies
58 like tacos
28 like pizza and hoagies
37 like hoagies and tacos
40 like pizza and tacos
25 like all three food

What is the number of students who like none or only one of the foods?

(A) 4
(B) 16
(C) 17
(D) 20
(E) 23

In a village of 100 households, 75 have at least one DVD player, 80 have at least one cell phone, and 55 have at least one MP3 player. If x and y are respectively the greatest and lowest possible number of households that have all three of these devices, x – y is:

(A) 65
(B) 55
(C) 45
(D) 35
(E) 25

You only have 100 households so there has to be overlap considering that the numbers, 75, 80, and 55 sum to 210. Let's look at the DVD Player and Cell Phone groups first because they are the biggest and will create the most obvious constraints. They sum to 155. Considering that there are only 100 households it must be that there are at least 55 overlaps, people who have both DVD and Cell. You could have the MP3 group completely contained in that 55. In that case, 55 people have three of the devices. That's the max that have all three of the devices.

Now let's try to minimize the people who have DVD, Cell Phone, and MP3 (I call these triples). With that in mind we want to make the MP3 people as independent as possible. There are 25 people who DON'T have a DVD (100 - 75) and 20 people who DON'T have a Cell Phone (100 - 80). That's 45 people total. Now, some of the 25 no DVD could be a part of the 20 Cell Phone but they don't have to be. The 20 and 25 could be independent and in this case we want those groups as unique as possible to maximize the number of people who are in ONLY MP3 player. Those 45 people could be ONLY MP3 player. So, because there are then ten leftover MP3 Player people (55 - 45) those 10 must be in all three groups. 55 (max in all three) - 10 (min in all three) = 45. Choose C.

You could also use the three group overlapping sets formulas:

Group A + Group B + Group C + None - Doubles - 2(Triples) = Total

210 - d - 2t = 100

-d - 2t = -110

d + 2t = 110

So now we know that that sum of d + 2t is 110. So the max t could be is 55 (2*55 = 110). A quick common sense check verifies that it is possible that all of the MP3 people are contained in the overlap between the DVD and Phone people. Now let's try to maximize the doubles. Can they be 110? Nope. You only have 100 households total. Can they be 100? Nope. Then you'd have 5 triples to balance out the equation. Again your max d + t is 100. Can they be 90? Then t is 10. That could work 90 + 10 = 100.

You can also visualize with Venn diagrams. For the most part I don't like Venns for three group (or two group) sets but for this one it can actually work relatively well. I'll circle back with a Venn Diagram solution.

# GMAT Question of the Day - Data Sufficiency - Overlapping Sets

Of the leopards at a certain zoo, 20% are both spotted and mature. If all the leopards at the zoo are either spotted or not spotted, or mature or immature, is the ratio of the number of immature leopards who are spotted to the number of immature leopards who are not spotted greater than 1?

(1) If the number of immature spotted leopards doubled than there would be 102 total leopards at the zoo.

(2) If the number of immature but not spotted leopards were decreased by a half there would 86 total leopards at the zoo.

## GMAT Question of the Day Solution

This overlapping sets question might leave you feeling a bit empty. Thinking - what was the point of that? Well, the point of that was to define the information that you were given and then to realize that you didn't have enough to answer the questions. That's what DS questions with an "E" answer are like. Sometimes there is no further meaning than just plain old insufficient.

## More GMAT overlapping sets practice questions!

Here's an overlapping sets question that adds in a bit of probability. Don't let the probability component get in your way! Often, probability questions are easy. Take it step by step.

Among a group of 2,500 people, 35 percent invest in municipal bonds, 18 percent invest in oil stocks, and 7 percent invest in both municipal bonds and oil stocks. If 1 person is to be randomly selected from the 2,500 people, what is the probability that the person selected will be one who invests in municipal bonds but NOT in oil stocks?

Here's a challenging sets question from the GMAT Prep Tests 1 and 2. Stay organized. Make the easy inferences and you should be fine.

In a certain city, 80 percent of the households have cable television, and 60 percent of the households have videocassette recorders. If there are 150,000 households in the city, then the number of households that have both cable television and videocassette recorders could be any number from:

Here's one from GMAT Question of the Day that's in the same vein. It also has percents/proportions and will give excellent practice using the two group overlapping sets matrix.

GMAT Question of the Day 2 group overlapping sets proportions and percents

This one is a bit different but will still give you 2 group practice and will give you exposure to overlapping sets on data sufficiency which often comes down to systems of questions/counting equations.

GMAT Question of the Day 2 group overlapping sets Data Sufficiency

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