GMAT Math : DSQ: Calculating discrete probability
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Example Questions
Example Question #3151 : Gmat Quantitative Reasoning
A test for a new drug was conducted. In the control or placebo group, 7 of 210 participants experienced positive results. The group that took the drug experienced 27 out of 374 positive resluts.
The placebo group had a sucess rate of .0333 and the drug group had a success rate of .0722. The difference is .0389 and the overall percentage (for both groups combined) is .0582
At what level is the difference of .0389 significant? Asked another way - what is the p-value for .0389?
The standard error of the difference (.0389) =
test statistic -
from the table (or excel NormsDist) - Z=1.9245 translates to .9729
Example Question #61 : Arithmetic
Jerry is a Cardinal fan and he and his family live on a street with 9 other families that are all Cardinal fans. One block to the north, there are 11 families and they are all Cub fans. These 21 households all buy their lawn fertilizer from Ben's Lawn and Garden Shop. Jerry suspects that Ben (who is originally from Chicago) is a Cub fan and that he provides better fertilizer to the Cub fans than to the Cardinal fans, while charging the same price for all.
Last Saturday everyone in town mowed their lawn. At 2:00 AM Sunday morning, Jerry snuck around town and weighed all of the grass clippings for the 21 households in question.
The weights (in lbs) of the grass clippings for the 10 Cardinal homes were:
82, 85, 90, 74, 80, 89, 75, 81, 93, 75
The weights (in lbs) of the grass clippings for the 11 Cub homes were:
90, 87, 93, 75, 88, 96, 90, 82, 95, 97, 78
The Cardinal average was 82.4; the Cub average was 88.27.
At what level is the 5.87 lb difference significant? - Asked another way - what is the p value for the 5.87 lb difference.
Cub variance = 53.218; Cardinal variance = 45.378.
The standard deviation of the difference (5.87) is:
TDIST of 1.918 with 19 Degrees of Freedom = .035
So, we would reject the null hypothesis (the hypothesis that claims that the means are equal) at 95% confidence (p=.05) and not reject at 99% (p=.01)
Example Question #62 : Arithmetic
A pair of dice - one fair, one loaded, but each with the usual numbers 1-6 on their faces - are rolled. What is the probability that one die will show a 5 and the other will show a 6?
Statement 1: The probability of rolling a 5 on the loaded die is 
Statement 2: The probability of rolling a 6 on the loaded die is 
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.
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.
BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.
The probability of rolling a 5 and the probability of rolling a 6 on the fair die are both 
Let the probabilities of rolling a 5 and a 6 on the loaded die be 
The probability of rolling a 6 on the fair die and a 5 on the loaded die is 
The probability of rolling a 5 on the fair die and a 6 on the loaded die is 
Therefore, the probability of rolling a 5-6 one way or the other is their sum:
which is dependent on both probabilites given in the two statements.
Example Question #63 : Arithmetic
Four cards are removed from a standard deck of fifty-two cards. Did the probability that a randomly drawn card will be a heart increase, decrease, or stay the same?
Statement 1: At least three of the cards are spades.
Statement 2: At least one of the cards is a club.
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 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.
BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.
One-fourth of the cards in a standard deck are hearts; in order to know whether the probability of drawing a heart increased, decreased, or stayed the same, we need to know whether fewer than, exactly, or more than one-fourth of the cards remaining are hearts. Therefore, we need to know whether no hearts, one heart, or more than one heart was removed.
Neither statement alone allows this question to be answered definitively, as we know nothing about the suit(s) of the other card(s). But both statements together tell us that no hearts were removed, so thirteen of the forty-eight cards remaining are hearts. This raises the probability from 
Example Question #64 : Arithmetic
Four cards are removed from a standard deck of fifty-two cards. Did the probability that a randomly drawn card will be a club increase, decrease, or stay the same?
Statement 1: The four cards that were removed were all kings.
Statement 2: One card of each suit was removed.
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.
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.
EITHER statement ALONE is sufficient to answer the question.
EITHER statement ALONE is sufficient to answer the question.
Thirteen of the fifty-two cards in a standard deck - one-fourth - are clubs, so the probability of drawing a club is 
Either statement tells us that one card of each suit was removed (there is one king of each suit), leaving twelve clubs out of forty-eight cards - one fourth of them. This keeps the probability of drawing a club at 
Example Question #61 : Arithmetic
A marble is selected at random from a box of red, yellow, and blue marbles. What is the probability that the marble is blue?
Statement 1: There are twice as many red marbles as yellow.
Statement 2: There are three times as many blue marbles as yellow.
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.
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.
BOTH statements TOGETHER are sufficient to answer the question, but NEITHER 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.
The first statement tells us the proportion of red marbles to yellow, and is helpful, but it tells us nothing about the number, relative or absolute, of blue marbles, so it alone does not answer the question; for similar reasons, neither does the second statement alone.
Together, however, they make the picture complete. If there are 
Example Question #66 : Arithmetic
A marble is selected at random from a box of red, yellow, and blue marbles. What is the probability that the marble is yellow?
Statement 1: Forty of the marbles are not red.
Statement 2: Sixty of the marbles are not blue.
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.
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.
BOTH statements TOGETHER are insufficient to answer the question.
We need to know the number of marbles that are yellow, and the number of marbles there are total. If we let 
These two statements together do not give us enough information. For example, these two situations both fit the conditions given:
Situation 1: 
Situation 2: 
Example Question #11 : Dsq: Calculating Discrete Probability
One card from another deck is added to a standard deck of fifty-two cards. The cards are shuffled and one card is removed.
A card is then drawn at random. What is the probability that that card is an ace?
Statement 1: The card that was added was a spade.
Statetment 2: The card that was removed was a jack.
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 sufficient to answer the question, but NEITHER statement ALONE is 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 insufficient to answer the question.
You need to know two things to answer this question - the rank of the added card, and the rank of the removed card. The second statement is useful but not sufficient; the first is irrelevant to the question.
Example Question #1051 : Data Sufficiency Questions
Two fair dice are rolled. What is the probability that each die (not the sum of the dice) will come up a composite number?
There are two composite numbers on each die, 4 and 6. On one die, the probability of rolling a 4 or a 6 is 
Example Question #1052 : Data Sufficiency Questions
Four coins - a fair penny, a loaded penny, a fair nickel, and a loaded nickel - are tossed. What is the probability that all four will come up heads?
Statement 1: Yesterday, out of 100 tosses, the loaded penny came up heads 70 times.
Statement 2: Yesterday, out of 100 tosses, the loaded nickel came up heads 40 times.
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 sufficient to answer the question, but NEITHER statement ALONE is 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.
BOTH statements TOGETHER are insufficient to answer the question.
While experiments such as repeated tossings can give an idea of the probability that a coin will come up heads or tails, they do not provide a definitive answer, so neither statement is helpful here.
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