### All GRE Math Resources

## Example Questions

### Example Question #11 : How To Find The Probability Of An Outcome

Mike has a bag of marbles. 5 are green, 8 are red, and 3 are blue. He pulls one marble out of the bag and it is green. He pulls out another one and it is red. He does not return these marbles to the bag. What is the probability that the next marble he pulls out of the bag will be green?

**Possible Answers:**

4/14

3/16

5/14

4/16

**Correct answer:**

4/14

First we need to find out how many marbles are in the bag in total. 5 + 8 + 3 = 16. He removes a green marble so now there are only 15 in total. When he removes the red one there are then 14 marbles in the bag. 14 is your denominator. The odds of picking a green one are 5 – 1 or 4 because there are only 4 green marbles left in the bag; therefore, the odds of picking another green marble is 4/14

### Example Question #12 : How To Find The Probability Of An Outcome

A classroom has 9 boys and 9 girls. One student is chosen at random as the class leader, and a second student is chosen at random as a back-up leader.

Quantity A: The probability of choosing a boy to be the leader and choosing a girl to be the back-up

Quantity B: The probability of choosing boys for both roles

**Possible Answers:**

Quantity A is greater

The relationship cannot be determined from the information give

The quantities are equal

Quantity B is greater

**Correct answer:**

Quantity A is greater

There are 18 students in the class, and 2 must be selected from the 9 girls and 9 boys. Key to this question is noting that the 2 students must be unique: ie once a student is selected to lead the class, he or she cannot be chosen to be the back-up.

Since these are independent events, the probability of each event is found, and the events are multiplied by each other to find the total.

Quantity A:

P(boy leader) = 9/18 = 1/2 as there are 9 boys out of a possible 18 students.

Once the boy has been chosen, there are 8 boys and 17 students total from which to choose the second student.

P(girl back-up) = 9/17 because there are 9 girls and 17 students left.

P(Quantity A) = (1/2)(9/17) = 9/34

Quantity B:

P(boy leader) = 9/18 = 1/2 as there are 9 boys out of a possible 18 students.

Once the boy has been chosen, there are 8 boys and 17 students total from which to choose the second student.

P(boy back-up) = 8/17 because there are 8 boys and 17 students left.

P(Quantity B) = (1/2)(8/17) = 8 / 34 = 4/17

### Example Question #13 : How To Find The Probability Of An Outcome

Aperture Industries is made up of 370 employees who work a total of 11,000 hours per week. If the number of weekly hours per employee has a normal distribution and standard deviation of 6 hours, approximately how many employees work more than 36 hours per week?

**Possible Answers:**

55

59

36

47

64

**Correct answer:**

59

First find the average number of hours worked by every employee: 11000/370 = 29.7 ~ 30 hrs/week.

Next, recognize that a single standard deviation encompasses 34% of the population on 1 end of the curve, or 68% of the population both above and below the mean. Since a standard deviation of 6 hours means that 68% of the population works between 24 to 36 hours per week, finding the amount that works over 36 hours is 100-68 = 32; 32/2 = 16% of the entire workforce on 1 end of the curve.

.16(370) = **59.2** or 59 employees work over 36 hrs/week.

### Example Question #14 : How To Find The Probability Of An Outcome

Quantitative Comparison

Max has 5 red marbles and 3 green marbles. He meets his friend Bob who has 4 white marbles and 4 green marbles. They combine them into a bag and pull individual marbles out one at a time with replacement.

Quantity A: The probability of picking a green marble from only Max's collection of marbles

Quantity B: The probability of picking a green marble from the combined bag of Max and Bob's marbles

**Possible Answers:**

The relationship cannot be determined from the information given.

Quantity B is greater.

Quantity A is greater.

The two quantities are equal.

**Correct answer:**

Quantity B is greater.

First let's evaluate Quantity A. Max has 5 red marbles and 3 green marbles, so the probability of picking a green marble is 3/8.

Next let's evaluate Quantity B. The bag of combined marbles now has 5 red, 7 green, and 4 white, so the probability of picking a green marble is 7/16.

To see which fraction is larger, we can make the two fractions have the same denominator. 3/8 is equivalent to 6/16, so now we can compare 6/16 to 7/16. Clearly Quantity B is bigger.

### Example Question #15 : How To Find The Probability Of An Outcome

At Jill's school fair, there is a game with 25 balloons hung on a dart board. 10 are blue, 8 are red, and 7 are green. Jill throws a dart and pops a blue balloon. What is the probability that the next balloon she hits will also be blue?

**Possible Answers:**

2/5

3/8

5/12

9/25

3/5

**Correct answer:**

3/8

Since one blue balloon has already been popped, there are now 9 blue balloons left, and 24 balloons left overall. Therefore the probability that the next balloon Jill hits is also blue is 9/24 = 3/8.

### Example Question #11 : How To Find The Probability Of An Outcome

The dealer gives you 4 cards from a regular 52-card deck, without replacement. What is the probability of receiving a spade, a heart, a diamond, and then a club, in that order?

**Possible Answers:**

(13/52) * (13/52) * (13/52) * (13/52)

(13/52) * (13/51) * (13/50) * (13/49)

(4/13) * (4/12) * (4/11) * (4/10)

(4/13) * (4/13) * (4/13) * (4/13)

(13/52) * (12/51) * (11/50) * (10/49)

**Correct answer:**

(13/52) * (13/51) * (13/50) * (13/49)

The probability of choosing a spade is 13/52. Now we have 51 cards to choose from, so the probability of then choosing a heart is 13/51. Now we have 50 cards left, with 12 spades, 12 hearts, 13 diamonds, and 13 clubs, so the probability of choosing a diamond is 13/50. Lastly we choose a club out of the remaining 49 cards with probability 13/49.

Therefore the answer is (13/52) * (13/51) * (13/50) * (13/49).

### Example Question #12 : How To Find The Probability Of An Outcome

There are 6 photos in a bag, numbered 1 to 6. The proportions of photos, *P*(*i*) for number *i*, are as follows: P(1) = 1/4, P(2) = 1/8, P(3) = 1/8, P(4) = 1/8, P(5) = 1/8, P(6) = 1/4. If a photo is drawn at random from the bag, what is the chance that the number on the photo is 3 or greater?

**Possible Answers:**

5/8

1/2

1/4

3/8

3/4

**Correct answer:**

5/8

Here we simply add up the proportions of the photos with numbers greater than or equal to 3.

P(3 or 4 or 5 or 6)

= P(3) + P(4) + P(5) + P(6)

= 1/8 + 1/8 + 1/8 + 1/4

= 5/8

### Example Question #18 : How To Find The Probability Of An Outcome

Three cars, *A*, *B*, and *C*, are in a race. *A* is twice as likely to win as *B*, and *B* is twice as likely to win as *C*. What is the probability that *B* *or* *C* wins?

**Possible Answers:**

2/3

3/7

3/4

1/2

1/7

**Correct answer:**

3/7

First let's find their individual probabilities of winning. Since *B* is twice as likely to win as *C*, P(*B*) = 2 * P(*C*). Since *A* is twice as likely to win as *B*, P(*A*) = 2 * P(*B*) = 2 * 2 * P(*C*) = 4 * P(*C*). We also know that the probabilities must sum to 1, so P(*A*) + P(*B*) + P(*C*) = 1, meaning P(*C*) + 2 * P(*C*) + 4 * P(*C*) = 1. Then P(*C*) = 1/7, so P(*B*) = 2 * P(*C*) = 2/7. Our answer is therefore P(*B* or *C*) = P(*B*) + P(*C*) = 2/7 + 1/7 = 3/7.

### Example Question #21 : Data Analysis

A singing group has 10 boys and 20 girls. Half the boys and half the girls have blue eyes. What is the probability that a student chosen at random from the group is a boy OR has blue eyes?

**Possible Answers:**

1/2

1/3

2/3

1/6

1/5

**Correct answer:**

2/3

P(boy) = 10/30 = 1/3

P(blue eyes) = 1/2

P(boy and blue eyes) = 5/30 = 1/6, because we are told that half (or 5) of the 10 boys have blue eyes

P(boy or blue eyes) = P(boy) + P(blue eyes) – P(boy and blue eyes)

= 1/3 + 1/2 – 1/6

= 2/3

### Example Question #20 : How To Find The Probability Of An Outcome

There are 31 students in the library.

Quantity A: The probability that all students have their birthdays in January. (There are 31 days in January).

Quantity B: The probability that all students have their birthdays on a Saturday this year. (Assume there are 52 Saturdays in a year).

**Possible Answers:**

The relationship cannot be determined from the information given.

Quantity B is greater.

The two quantities are equal.

Quantity A is greater.

**Correct answer:**

Quantity B is greater.

As the options suggest, this problem requires us to know how probabilities work. The probability of a single event happening is equal to the number of ways it can happen divided by the total number of outcomes. Because this problem asks about 31 students, we have to raise our probability to the 31st power because it's the probability of 31 different events happening.

Now we can figure out the probability of one January birthday (31 days in January/365 days in a year) and the probability of all Saturday birthdays (52 Saturdays in a year/365 days in a year). So thus the probabilities that ALL of the students will have either type of birthday is:

(31/365)^{31} and

(52/365)^{31}

Since the numbers are raised to the same power, we can simply look at the base to determine which is larger and which is smaller. Since the probability of having a birthday on Saturday (roughly 1/7) is greater than having a birthday in January (roughly 1/12), then Quantity B is greater.