This question tests middle school quantitative reasoning skills, specifically solving basic probability problems. Probability measures the likelihood of an event occurring, calculated as the ratio of favorable outcomes to the total number of possible outcomes. In this scenario, students are asked to determine the probability of rolling a sum of 7 with two dice based on the sample space of 36 possible outcomes. Choice D is correct because it accurately calculates the probability as 1/6, using the 6 favorable outcomes for sum 7 out of 36 total rolls. Choice B is incorrect because it counts only one specific pair, leading to 1/36. This error often occurs when students overlook multiple ways to achieve the sum. To help students, encourage them to carefully list all possible outcomes and use clear diagrams or tables to visualize probabilities. Practice converting between fractions, decimals, and percentages, and emphasize checking calculations for accuracy.

First, find the number of multiples of 7 from 1 to 100. This is \(\lfloor \frac{100}{7} \rfloor = 14\). Next, we need to exclude the numbers that are also multiples of 5. A number that is a multiple of both 7 and 5 is a multiple of their least common multiple, which is 35. The multiples of 35 from 1 to 100 are 35 and 70. There are 2 such numbers. The number of pages that are multiples of 7 but not 5 is \(14 - 2 = 12\). The total number of pages is 100. So, the probability is \(\frac{12}{100} = \frac{3}{25}\).

First, find the probability of selecting a green marble. The sum of the probabilities for all outcomes must be 1. The probability of selecting a red or blue marble is \(\frac{1}{4} + \frac{1}{3} = \frac{3}{12} + \frac{4}{12} = \frac{7}{12}\). Therefore, the probability of selecting a green marble is \(1 - \frac{7}{12} = \frac{5}{12}\). Let T be the total number of marbles. We know that the number of green marbles is 10, so \(\frac{5}{12} \times T = 10\). To find T, we can multiply both sides by \(\frac{12}{5}\): \(T = 10 \times \frac{12}{5} = \frac{120}{5} = 24\). The total number of marbles is 24.

The set of integers from -5 to 5, inclusive, is {-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5}. To find the total number of integers, we calculate \(5 - (-5) + 1 = 11\). The numbers in this set that are positive and even are {2, 4}. There are 2 favorable outcomes. Therefore, the probability is \(\frac{2}{11}\).

Initially, there are 20 balls with a red to blue ratio of 3:2. This means there are \(3+2=5\) parts. The value of one part is \(20 \div 5 = 4\). So, there are \(3 \times 4 = 12\) red balls and \(2 \times 4 = 8\) blue balls. Then, 4 red balls and 1 blue ball are added. The new number of red balls is \(12 + 4 = 16\). The new number of blue balls is \(8 + 1 = 9\). The new total number of balls is \(20 + 4 + 1 = 25\). The new probability of picking a blue ball is \(\frac{\text{new number of blue balls}}{\text{new total number of balls}} = \frac{9}{25}\).

This question tests middle school quantitative reasoning skills, specifically solving basic probability problems. Probability measures the likelihood of an event occurring, calculated as the ratio of favorable outcomes to the total number of possible outcomes. In this scenario, students are asked to determine the probability of drawing a heart from a 52-card deck based on the sample space of 52 cards. Choice C is correct because it accurately calculates the probability as 1/4, using the 13 hearts out of 52 cards. Choice A is incorrect because it counts only one specific heart, leading to 1/13. This error often occurs when students overlook the total number of cards in a suit. To help students, encourage them to carefully list all possible outcomes and use clear diagrams or tables to visualize probabilities. Practice converting between fractions, decimals, and percentages, and emphasize checking calculations for accuracy.

This question tests middle school quantitative reasoning skills, specifically solving basic probability problems. Probability measures the likelihood of an event occurring, calculated as the ratio of favorable outcomes to the total number of possible outcomes. In this scenario, students are asked to determine the probability of getting at least one head when flipping two coins based on the sample space of 4 outcomes. Choice B is correct because it accurately calculates the probability as 3/4, using the 3 favorable outcomes out of 4 total flips. Choice A is incorrect because it calculates the probability of both tails, leading to 1/4. This error often occurs when students overlook complementary counting. To help students, encourage them to carefully list all possible outcomes and use clear diagrams or tables to visualize probabilities. Practice converting between fractions, decimals, and percentages, and emphasize checking calculations for accuracy.

This question tests middle school quantitative reasoning skills, specifically solving basic probability problems. Probability measures the likelihood of an event occurring, calculated as the ratio of favorable outcomes to the total number of possible outcomes. In this scenario, students are asked to determine the probability of drawing a heart from a shuffled deck based on the sample space of 52 cards. Choice B is correct because it accurately calculates the probability as 1/4, using the 13 hearts out of 52 cards. Choice A is incorrect because it assumes half the deck are hearts, leading to 1/2. This error often occurs when students overlook the four suits. To help students, encourage them to carefully list all possible outcomes and use clear diagrams or tables to visualize probabilities. Practice converting between fractions, decimals, and percentages, and emphasize checking calculations for accuracy.

This question tests middle school quantitative reasoning skills, specifically solving basic probability problems. Probability measures the likelihood of an event occurring, calculated as the ratio of favorable outcomes to the total number of possible outcomes. In this scenario, students are asked to determine the probability of drawing a red marble from a bag with 2 red, 6 blue, and 2 green marbles based on the sample space of 10 marbles. Choice D is correct because it accurately calculates the probability as 2/10, using the 2 red marbles out of 10 total. Choice A is incorrect because it assumes half are red, leading to 1/2. This error often occurs when students overlook the actual counts. To help students, encourage them to carefully list all possible outcomes and use clear diagrams or tables to visualize probabilities. Practice converting between fractions, decimals, and percentages, and emphasize checking calculations for accuracy.

This question tests middle school quantitative reasoning skills, specifically solving basic probability problems. Probability measures the likelihood of an event occurring, calculated as the ratio of favorable outcomes to the total number of possible outcomes. In this scenario, students are asked to determine the probability of drawing a red marble from a bag with 5 red, 3 blue, and 2 green marbles based on the sample space of 10 marbles. Choice A is correct because it accurately calculates the probability as 1/2, using the 5 red marbles out of 10 total. Choice C is incorrect because it uses the blue marbles instead, leading to 3/10. This error often occurs when students overlook the color specified in the question. To help students, encourage them to carefully list all possible outcomes and use clear diagrams or tables to visualize probabilities. Practice converting between fractions, decimals, and percentages, and emphasize checking calculations for accuracy.