GMAT Math : Discrete Probability
Study concepts, example questions & explanations for GMAT Math
All GMAT Math Resources
Example Questions
Example Question #82 : Dsq: Calculating Discrete Probability
Some balls are placed in a large box; the balls include one ball marked "10", two balls marked "9", and so forth up to ten balls marked "1". A ball is drawn at random.
Statement 1: 
Statement 2:
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.
STATEMENT 1 ALONE provides sufficient information to answer the question, but STATEMENT 2 ALONE does NOT provide sufficient information to answer the question.
EITHER STATEMENT ALONE provides 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.
STATEMENT 2 ALONE provides sufficient information to answer the question, but STATEMENT 1 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 frequencies out of 55 of each outcome from 1 to 10, in order, are as follows:
Their respective probabilities are their frequencies divided by 55:
The probability that the ball will be marked "5" is
therefore, the probability that the ball will be marked with any given integer less than or equal to 5 will be greater than 
The probability that the ball will be marked "6" is
therefore, the probability that the ball will be marked with any given integer greater than or equal to 6 will be less than 
Therefore, it suffices to know whether the number on the ball is less than or equal to 5. Statement 2 states that the number on the ball is less than or equal to 5, so it is sufficient to answer the question in the affirmative. Statement 1 is insufficient, since there are primes less than or equal to 5 - 2, 3, and 5 - and one prime greater than 5, which is 7.
Example Question #83 : Dsq: Calculating Discrete Probability
Some balls are placed in a large box; the balls include one ball marked "10", two balls marked "9", and so forth up to ten balls marked "1". A ball is drawn at random.
Statement 1: 
Statement 2: 
STATEMENT 1 ALONE provides sufficient information to answer the question, but STATEMENT 2 ALONE does NOT provide sufficient information to answer the question.
EITHER STATEMENT ALONE provides 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.
BOTH STATEMENTS TOGETHER provide sufficient information to answer the question, but NEITHER STATEMENT ALONE provides 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.
The total number of balls in the box will be
Since
it follows that the number of balls is
The frequencies out of 55 of each outcome from 1 to 10, in order, is as follows:
Their respective probabilities are their frequencies divided by 55:
The probability that the ball will be marked "5" is
therefore, the probability that the ball will be marked with any given integer less than or equal to 5 will be greater than
The probability that the ball will be marked "6" is
therefore, the probability that the ball will be marked with any given integer greater than or equal to 6 will be less than
Therefore, it suffices to know whether the number on the ball is less than or equal to 5.
Statement 1 alone is insufficient, since there are two perfect square integers from 1 to 5 (1 and 4) and one perfect square integer from 6 to 10 (9). Statement 2 alone is insufficient, since it is not clear whether the number on the ball is 5 or a number greater than 5. However, from the two statements together, it can be inferred that 
Example Question #3221 : Gmat Quantitative Reasoning
A bag contains x red marbles, y blue marbles, and z green marbles. What is the probability of drawing a green marble?
(1) There's a 
(2) There's a 
EACH statement ALONE is sufficient.
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
Statements (1) and (2) TOGETHER are NOT sufficient.
BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
Probability is a part to whole comparison.
With statement 1, we don't know nor can we determine the probability of drawing a green marble because we know nothing about the probability of drawing a blue marble. Therefore, statement 1 is not sufficient.
With statement 2, we don't know nor can we determine anything regarding the probability of drawing a red marble. Therefore, statement 2 is not sufficient.
However, taken together, we can determine that the probability of drawing a green marble is 
Example Question #131 : Arithmetic
A coin is tossed n times. What is the probability of getting at least one tail?
(1) The probability of never flipping a head is 
(2) The probability of flipping at least one head is 
BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
EACH statement ALONE is sufficient.
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
Statements (1) and (2) TOGETHER are NOT sufficient.
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
EACH statement ALONE is sufficient.
Probability is the number of desired outcomes divided by the number of possible outcomes.
With statement 1, we are told the probability of never flipping a head is 
Therefore, statement 1 alone is sufficient.
With statement 2, we are told the probability of flipping at least one head is 
Therefore, the correct answer is EACH statement ALONE is sufficient.
Example Question #131 : Arithmetic
Several decks of playing cards are shuffled together. One card is drawn, shown, and put aside. Another card is dealt. What is the probability that the dealt card is red, assuming the first card is known?
1) The card removed before the deal was red.
2) The cards were shuffled again between the draw and the deal.
BOTH statements TOGETHER are insufficient to answer the question.
Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is not sufficient.
BOTH statements TOGETHER are sufficient to answer the question.
EITHER statement ALONE is sufficient to answer the question.
Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is not sufficient.
BOTH statements TOGETHER are insufficient to answer the question.
To answer this question you need to know two things: the number of red cards left and the number of total cards left. The second statement is irrelevant, as a reshuffle does not change the composition of the deck. The first statement tells you that there is one fewer red card than black cards, but it does not tell you how many of each there are, as you do not know how many decks of cards there were.
And that information, which is not given, affects the answer. For example, if there were four decks, there were 103 red cards out of 207; if there were six decks, there were 155 red cards out of 311. The probabilities would be, respectively,
and
a small difference, but nonetheless, a difference.
The correct answer is that both statements together are insufficient to answer the question.
All GMAT Math Resources
