All GMAT Math Resources
Example Questions
Example Question #75 : Calculating Discrete Probability
Some balls are placed in a large box; the balls include one ball marked "A", two balls marked "B", and so forth up to twenty-six balls marked "Z". A ball is drawn at random.
Given a particular letter of the alphabet, does the probability that that ball will be marked with that letter exceed ?
Statement 1: The letter appears in the word "lousy".
Statement 2: The letter appears in the word "skunk".
BOTH STATEMENTS TOGETHER provide sufficient information to answer the question, but NEITHER STATEMENT ALONE provides sufficient information to answer the question.
EITHER STATEMENT ALONE provides sufficient information to answer the question.
STATEMENT 1 ALONE provides sufficient information to answer the question, but STATEMENT 2 ALONE does NOT provide sufficient information to answer the question.
STATEMENT 2 ALONE provides sufficient information to answer the question, but STATEMENT 1 ALONE does NOT provide sufficient information to answer the question.
BOTH STATEMENTS TOGETHER do NOT provide sufficient information to answer the question.
STATEMENT 1 ALONE provides sufficient information to answer the question, but STATEMENT 2 ALONE does NOT provide sufficient information to answer the question.
The total number of balls in the box will be
.
Since
,
it follows that the number of balls is
.
The number of balls with a given letter of the alphabet is equal to the number of its position in the alphabet; the probability of a ball with that letter being drawn is that number divided by the total number of balls, 351. Therefore, for this probability to exceed , we must have the relation
.
Therefore, .
The 11th letter of the alphabet is "K", so in order to answer this question, it suffices to know whether the letter comes after "K" in the alphabet.
From Statement 1 alone, the question can be answered, since all of the letters in the word "lousy" appear after "K" in the alphabet. From Statement 2 alone, however, the question cannot be answered, since the letter "K" itself appears in the word "skunk".
Example Question #71 : Calculating Discrete Probability
The upper portion of the above spinner has radius twice that of the lower portion.
If the above spinner is spun, what are the odds against the arrow stopping in the purple region?
11 to 1
11 to 2
7 to 1
13 to 2
6 to 1
11 to 1
The radii of the portions of the spinner - and the areas of the sectors - are actually irrelevant to the problem; it is the measures of their central angles that count.
The purple sector is one-sixth of the larger semicircle, so it is one-twelfth of a circle. This means that the probability of a spinner stopping inside that sector is , and the odds against this are
- that is, 11 to 1.
Example Question #77 : Calculating Discrete Probability
Some balls are placed in a large box; the balls include one ball marked "A", two balls marked "B", and so forth up to twenty-six balls marked "Z". A ball is drawn at random.
Given a particular letter of the alphabet, does the probability that that ball will be marked with that letter exceed ?
Statement 1: The letter in question is in the word "pique".
Statement 2: The letter in question is a vowel.
(Note: for purposes of this problem, "Y" is a consonant.)
STATEMENT 1 ALONE provides sufficient information to answer the question, but STATEMENT 2 ALONE does NOT provide sufficient information to answer the question.
STATEMENT 2 ALONE provides sufficient information to answer the question, but STATEMENT 1 ALONE does NOT provide sufficient information to answer the question.
EITHER STATEMENT ALONE provides sufficient information to answer the question.
BOTH STATEMENTS TOGETHER do NOT provide sufficient information to answer the question.
BOTH STATEMENTS TOGETHER provide sufficient information to answer the question, but NEITHER STATEMENT ALONE provides sufficient information to answer the question.
BOTH STATEMENTS TOGETHER do NOT provide sufficient information to answer the question.
The total number of balls in the box will be
.
Since
,
it follows that the number of balls is
.
The number of balls with a given letter of the alphabet is equal to the number of its position in the alphabet; the probability of a ball with that letter being drawn is that number divided by the total number of balls, 351. Therefore, for this probability to exceed , we must have the relation
.
Therefore, .
The 17th letter of the alphabet is "Q", so in order to answer this question, it suffices to know whether the letter comes after "Q" in the alphabet.
Assume both statements to be true. There are three vowels in "pique"; "E" and "I" come before "Q" and "U", after "Q". Therefore, the two statements together are inconclusive.
Example Question #111 : Outcomes
Aaron, Gary, Craig, and Boone are sitting down in a row of four chairs. What is the probability that Aaron and Gary will be seated beside each other?
Consider first all of the possible ways the men may be arranged, which is
Now, consider all of the ways that Aaron and Gary could be seated beside each other; it may be easier to visualize by drawing it out:
- A G _ _
- G A _ _
- _ A G _
- _ G A _
- _ _ A G
- _ _ G A
As seen, there are six possibilities.
Finally, for each of these cases, Craige and Boone could be seated in one of two ways.
So the probability that Aaron and Gary will be seated beside each other is:
Example Question #1 : Counting Methods
In how many ways can the 11th grade class elect a president, vice president, and treasurer from a class of 70 students?
The president can be elected in 70 different ways. After a student is elected president, there are 69 students left to elect a vice president from. Similarly, there are then 68 students left for the spot of treasurer. So there are different arrangements.
Example Question #1 : Understanding Counting Methods
What is the number of possible 4 letter code words that can be made from the alphabet, when all 4 letters must be different?
4!
26!
22!
This is a permutation of 26 objects (letters) taken 4 at a time. Here order matters, because for example, "abcd" is not the same code word as "bdca".
You must know the permutation formula! It is as follows:
, where n is the number of different objects taken r at a time.
Here we have
Note: This is equivalent to 26 * 25 * 24 * 23.
Example Question #2 : Counting Methods
There are 8 paths between places and and 5 paths betweeen places and . How many different routes are there between places and ?
Multiple the number of routes for each piece of the trip:
Example Question #4 : Understanding Counting Methods
How many subsets does a set with 12 elements have?
The number of subsets in a set of size is . If , then the set has subsets.
Alternatively, each subset of this twelve-element set is essentially a sequence of 12 independent decisions, one per element - each decision has two possible outcomes, exclusion or inclusion. By the multiplication principle, this is 2 taken as a factor 12 times, or
Example Question #3 : Counting Methods
How many ways can a president, a vice-president, a secretary-treasurer, and three Student Senate representatives be selected from a class of thirty people? You may assume these will be six different people.
This can be seen, without loss of generality, as choosing each officer in turn.
There are 30 ways of choosing the president; there are then 29 ways of choosing the vice-president, and 28 ways of choosing the secretary-treasurer. Then 3 Student Senate representatives are chosen from the remaining 27 students; this is a combination of 3 elements from 27 - that is, . By the multiplication principle, the number of possible selections of the officers is:
Example Question #1871 : Problem Solving Questions
How many ways can you select three different prime numbers between 1 and 20?
There are eight prime numbers between 1 and 20:
The number of ways to select three of them, without regard to order, is the number of combinations of three out of eight: