GMAT Math : Arithmetic
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Example Questions
Example Question #51 : Discrete Probability
Jenny altered a coin. Did the probability that it would come up heads increase, decrease, or stay the same?
Statement 1: Jenny flipped the coin 100 times; it came up heads 55 times.
Statement 2: Her friend Susie then flipped the coin 1,000 times and it came up heads 557 times.
EITHER 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.
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.
BOTH statements TOGETHER are insufficient to answer the question.
BOTH statements TOGETHER are insufficient to answer the question.
The two statements together are insufficient.
Jenny's and Susie's experiments both yield empirical data, which is not in and of itself sufficient to draw a definite conclusion about the theoretical probability of any outcomes. While their results strongly suggest that the probability of the coin coming up heads has increased, it is entirely possible for a fair coin, or even a coin altered to come up tails more often, to be flipped with these results as well.
Example Question #102 : Arithmetic
Xenia altered a die. Did the probability that it would come up a 6 increase, decrease, or stay the same?
Statement 1: The probability that the sum of two rolls of the die will be a "11" decreased.
Statement 2: The probability that the sum of two rolls of the die will be "12" decreased.
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.
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.
Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.
The probability that a roll of an unaltered die will yield an outcome of "6" is 
Assume Statement 1 alone. There is only one way to roll a die twice so that the sum of the rolls comes up a "11" - a"5" and a "6", in either order. The probability of rolling a "5" and a "6" on the altered die in that order is 
On an unaltered die, the probability of this happening would be 
By Statement 1, the alterations decreased this probability, so
and
Without further information, it cannot be determined whether 
For example, if 
If 
Both scenarios fit the conditions of Statement 1, but in one, the probability of rolling a "6" on the die remained the same, and in the other, the probability decreased. Statement 1 gives insufficient information.
Assume Statement 2 alone. There is only one way to roll a die twice so that the sum of the rolls comes up a "12" - a double "6". The probability doing this twice on the altered die is 
Xenia's alterations decreased the probability of rolling a "6" on the die.
Example Question #103 : Arithmetic
Quentin altered a die. Did the probability that it would come up a 6 increase, decrease, or stay the same?
Statement 1: Quentin rolled the die 120 times; it came up a "6" 20 times.
Statement 2: His friend Allen then rolled the die 600 times; it came up a "6" 100 times.
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.
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.
BOTH statements TOGETHER are insufficient to answer the question.
The two statements together are insufficient.
Quentin's and Allen's experiments both yield empirical data, which is not in and of itself sufficient to draw a definite conclusion about the theoretical probability of any outcomes. While their results strongly suggest that the probability of the die coming up a "6" remained unchanged (since 
Example Question #104 : Arithmetic
Roger 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 red card - did it change or did it stay the same?
Statement 1: The card Roger took out of the deck was a heart.
Statement 2: Roger ran 100 experiments using the modified deck, each involving a shuffle and a draw; he drew a red card 48 times.
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.
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 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT 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.
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 red cards was changed - that is, if the card Roger replaced with the joker was a red card.
Statement 2 alone provides this information. Statement 1 is unhelpful, in that it deals with an empirical result. It is entirely possible for this to happen in an unmodified deck, or even in a deck with more black cards than red.
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.
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