GMAT Math : GMAT Quantitative Reasoning
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Example Questions
Example Question #105 : Arithmetic
Brandy altered a coin. Did the probability of a toss of the coin coming up heads increase or decrease?
Statement 1: The probability that, if the coin is tossed four times, all four tosses will result in tails decreased.
Statement 2: The probability that, if the coin is tossed six times, all six tosses will result in tails decreased.
BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.
Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.
EITHER statement ALONE is sufficient to answer the question.
BOTH statements TOGETHER are insufficient to answer the question.
Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.
EITHER statement ALONE is sufficient to answer the question.
Let 
Assume Statement 1 alone.
The probability of a single toss of a fair coin coming up heads (or tails) is 
The probability of four tails in a row on the altered coing will be 
The probability of one toss of the coin coming up tails decreased, so the probability of it coming up heads increased.
A similar argument can be used to demonstrate that Statement 2 allows the same conclusion to be drawn.
Example Question #1083 : Data Sufficiency Questions
Dick replaced a card in a standard 52-card deck with the joker. What happened to the probability that a randomly-drawn card would be a black card - did it change or did it stay the same?
Statement 1: Dick did not look at the card he took out of the deck.
Statement 2: The card Dick took out of the deck was a spade.
BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is 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 to answer the question.
Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.
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 to answer the question.
Since the number of cards in the deck remained unchanged, the probability of a random draw resulting in a black card changed if and only if the number of black cards was changed - that is, if the card Dick replaced with the joker was a black card. Statement 2, but not Statement 1, provides this information. Note that Statement 1 is irrelevant - whether Dick knew the color of the card does not change any theoretical probability.
Example Question #107 : Arithmetic
Jeremiah has a bag of coins. What are the odds of him pulling a 
I) There are three types of coins in the bag; 
II) There are twice as many 
Neither statement is sufficient to answer the question. More information is needed.
Both statements are needed to answer the question.
Statement II is sufficient to answer the question, but statement I is not sufficient to answer the question.
Statement I is sufficient to answer the question, but statement II is not sufficient to answer the question.
Either statement is sufficient to answer the question.
Neither statement is sufficient to answer the question. More information is needed.
In order to calculate probability, we need to know the total number of coins, as well as the number of each type of coin.
In this case, I) tells us how many types of coins there are, and II) gives us a clue about the ratio of 5 euro to 50 cent coins.
The key information we are missing is the amount of 2 euro coins that are in the bag. Without this piece of information neither statement will allow us to answer the question.
Therefore, we do not have enough information to answer this question.
Example Question #108 : Arithmetic
In a standard deck of cards there are 
If the following are true, what are the odds of pulling a diamond followed by a club?
I) This deck contains two jokers with no suit.
II) Three spades have been removed from the deck.
Either statement is sufficient to answer the question.
Both statements are needed to answer the question.
Statement II is sufficient to answer the question, but statement I is not sufficient to answer the question.
Statement I is sufficient to answer the question, but statement II is not sufficient to answer the question.
Neither statement is sufficient to answer the question. More information is needed.
Both statements are needed to answer the question.
When calculating probability we need to know the total amount of cards we are pulling from and then also, how many cards from the total are diamonds and how many are clubs.
Since we are given the total amount of cards and the amount cards of each suit from the original question, we see that both statements are needed to calculate the final probability because each statement changes the makeup of our original deck.
I) Gives us an additional two cards to include in our probability calculations.
II) Takes two cards away from our calculations.
Therefore, both are necessary to calculate the correct probability.
Example Question #52 : Discrete Probability
The state fair is holding a raffle for a prize. 500 tickets are put up for sale. What is the expected value of each ticket, assuming all tickets are sold?
Statement 1: Each ticket is sold for $5.
Statement 2: The prize is a new laptop computer.
Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.
EITHER statement ALONE is sufficient to answer the question.
BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.
Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.
BOTH statements TOGETHER are insufficient to answer the question.
BOTH statements TOGETHER are insufficient to answer the question.
Assume both statements are true.
Three things are needed to determine expected value of a ticket:
The price of the ticket, which is given in Statement 1;
The number of tickets sold, which is given in the main body of the problem; and,
The worth of the prize, which is not given anywhere (Statement 2 only identifies the prize; it does not give its value).
The two statements together provide insufficient information.
Example Question #110 : Arithmetic
A state fair is holding a raffle for a laptop computer; there are no other prizes. Ted wants to buy a ticket, but only if its expected value is greater than 
Statement 1: The laptop has value $800.
Statement 2: The people running the state fair will sell 400 tickets for $5 each.
Note: You may assume all tickets are sold.
Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.
BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.
Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.
EITHER statement ALONE is sufficient to answer the question.
BOTH statements TOGETHER are insufficient to answer the question.
BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.
Three things are needed to determine the expected value of a game - the probability of a win, the value of a win, and the value of a loss.
Statement 1 provides insufficient information. The statement does not give the number of tickets sold, so no clue as to the probability of a win is given. Also, neither the value of a win not that of a loss is given; the worth of the prize is given, but without the price of the ticket, the value of the win (the worth of the prize minus the price of a ticket) cannot be determined.
Statement 2 provides insufficient information. The statement gives the value of a loss: 
Assume both statements hold. The value of a win is $795—the $800 value of the laptop minus the $5 ticket price. The value of a loss is 
which justifies Ted purchasing a ticket under his self-imposed criterion.
Example Question #3201 : Gmat Quantitative Reasoning
What is the probability that a patient admitted to a certain hospital will have 2 tests performed upon entering the hospital?
(1) A study conducted in the hospital revealed that 
(2) A study conducted in the hospital revealed that 1 in 5 patients will have 2 tests performed.
Each Statement ALONE is sufficient.
Statements (1) and (2) TOGETHER are not sufficient.
Both statements TOGETHER are sufficient, but NEITHER 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
Statement (2) ALONE is sufficient, but Statement (1) ALONE is not sufficient
(1) A study conducted in the hospital revealed that
Using the information in Statement (1), we cannot find the probability of patients who have exactly two tests performed upon entering the hospital. The probability given in Statement (1) also includes patients who have more than 2 tests performed.
Therefore, Statement (1) Alone is not sufficient.
(2) A study conducted in the hospital revealed that 1 in 5 patients will have 2 tests performed.
Using Statement (2), we can calculate the probability that a patient entering the hospital will have exactly two tests performed as:
Example Question #112 : Arithmetic
What is the probability that a student guesses the correct answer to each question of a quiz?
(1) There are only 3 questions on the quiz.
(2) Each question has 4 answer choices and only one choice is correct.
Each Statement ALONE is sufficient.
Statements (1) and (2) TOGETHER are not sufficient.
Both statements TOGETHER are sufficient, but NEITHER 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.
Both statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
(1) There are only 3 questions on the quiz.
Using Statement (1), we know that the probability that the student guesses the correct answer to each question is the product of the probabilities of getting each individual question right (getting each question right constitutes independent events). That is:
Pr(Question1 Correct) x Pr(Question2 Correct) x Pr(Question3 Correct)
However Statement (1) Alone is not sufficient.
(2) Each question has 4 answer choices and only one choice is correct.
Using Statement (2), we know that the student has the same probability of getting each question right, which we calculate as:
Pr(Correct Answer)=
Still Statement (2) Alone is not Correct.
Combining Both statements, we find the probability of guessing each correct answer on the quiz as:
Therefore Both statements Together are sufficient but neither Statement Alone is sufficient.
Example Question #3201 : Gmat Quantitative Reasoning
Bronson has a box of 16 markers. The markers are green, red and yellow.
I) The number of green markers is twice the number of red markers
II) There are 4 yellow markers
What are the odds of pulling a yellow marker followed by a green marker followed by a red marker? (Assume no replacement.)
Neither statement is sufficient to answer the question. More information is needed.
Statement II is sufficient to answer the question, but Statement I is not sufficient to answer the question.
Either statement is sufficient to answer the question.
Statement I is sufficient to answer the question, but Statement II is not sufficient to answer the question.
Both statements are needed to answer the question.
Both statements are needed to answer the question.
To find probability, we need to know the number of each color of marker. Statement I gives us a way to put the number of red markers in terms of green markers. Statement II tells us the number of yellow markers. Using Statement II, we can find out how many red and green markers we have together. Then we would have to use Statement I to find the number of red markers and green markers. Then we can calculate probability. There is no way to do it without both statements.
To recap:
Bronson has a box of 16 markers. The markers are green, red and yellow.
I) The number of green markers is twice the number of red markers
II) There are 4 yellow markers
What are the odds of pulling a yellow marker followed by a green marker followed by a red marker? (Assume no replacement.)
So:
So, we have 4 yellow markers, 4 red markers and 8 yellow markers.
To find the odds of three events, we need to multiply the probability of each event together. We assume no replacement, so for each event, our total number of outcomes decreases by one.
So, the odds are 4 in 105.
Example Question #1092 : Data Sufficiency Questions
Two fair (but possibly, but not necessarily, altered) six-sided dice are tossed, and their sum is noted. Below is a probability table for all possible outcomes.
Which of these choices is a possibility for the numbers on the faces of the dice?
According to the probability table, only an odd number can be rolled. Three of our choices can be eliminated immediately for this reason.
If the dice have 
This leaves two possibilites. All we need to look at is the probability of rolling a 3 with each choice.
A 3 can only be rolled by rolling a 1 on the first die and a 2 on the second. The probability of doing this is
Again, a 3 can only be rolled by rolling a 1 on the first die and a 2 on the second. The probability of doing this is
The other probabilities can be confirmed to agree with the ones given in the conditions in the problem. This is the correct choice.
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