A single card is drawn from a standard 52-card deck. What is the probability that the card is a heart or a face card (Jack, Queen, or King)?
Opening subject page...
Loading your content
ISEE Middle Level Mathematics Achievement Quiz
Practice Calculating Probability in ISEE Middle Level Mathematics Achievement with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.
Question 1 / 20
0 of 20 answered
A single card is drawn from a standard 52-card deck. What is the probability that the card is a heart or a face card (Jack, Queen, or King)?
This quiz focuses on Calculating Probability, giving you a quick way to practice the rules, question types, and explanations that matter most for ISEE Middle Level Mathematics Achievement.
Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.
A single card is drawn from a standard 52-card deck. What is the probability that the card is a heart or a face card (Jack, Queen, or King)?
Explanation: There are 13 hearts in a 52-card deck. There are 12 face cards (3 in each of the 4 suits). However, the Jack, Queen, and King of hearts are both hearts and face cards. To avoid double-counting, we use the formula (P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)). The number of favorable outcomes is the number of hearts plus the number of face cards minus the number of cards that are both: (13 + 12 - 3 = 22). The total number of possible outcomes is 52. Therefore, the probability is (\frac{22}{52}), which simplifies to (\frac{11}{26}).
A number is randomly chosen from the set of all factors of 72. What is the probability that the chosen number is a multiple of 3?
Explanation: First, list all the factors of 72. The factors are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72. There are a total of 12 factors. Next, identify which of these factors are multiples of 3. The multiples of 3 in this list are 3, 6, 9, 12, 18, 24, 36, 72. There are 8 such numbers. The probability is the ratio of the number of favorable outcomes to the total number of outcomes: (\frac{8}{12} = \frac{2}{3}).
A jar contains 5 red marbles, 4 blue marbles, and 3 green marbles. If two marbles are drawn from the jar at random without replacement, what is the probability that both marbles are blue?
Explanation: The total number of marbles in the jar is (5 + 4 + 3 = 12). The probability of the first marble being blue is (\frac{4}{12}). After one blue marble is drawn, there are 11 marbles left, and 3 of them are blue. So, the probability of the second marble being blue is (\frac{3}{11}). To find the probability of both events happening, multiply their probabilities: (\frac{4}{12} \times \frac{3}{11} = \frac{1}{3} \times \frac{3}{11} = \frac{1}{11}).
The probability that event A occurs is 0.4, and the probability that event B occurs is 0.5. If the probability that both A and B occur is 0.1, what is the probability that neither A nor B occurs?
Explanation: The probability that either A or B (or both) occurs is given by the formula (P(A \cup B) = P(A) + P(B) - P(A \cap B)). Using the given values, (P(A \cup B) = 0.4 + 0.5 - 0.1 = 0.8). The event that 'neither A nor B occurs' is the complement of the event that 'either A or B occurs'. Therefore, the probability is (1 - P(A \cup B) = 1 - 0.8 = 0.2).
An integer is randomly selected from the integers 1 to 40, inclusive. What is the probability that the selected integer is a multiple of 4 or a multiple of 6?
Explanation: There are 40 integers in total. The number of multiples of 4 is (40 \div 4 = 10). The number of multiples of 6 is (40 \div 6 = 6) with a remainder (the multiples are 6, 12, 18, 24, 30, 36). To find the number of integers that are multiples of 4 or 6, we add the counts and subtract the overlap. The overlap consists of multiples of the least common multiple of 4 and 6, which is 12. The multiples of 12 up to 40 are 12, 24, and 36 (3 multiples). The number of favorable outcomes is (10 + 6 - 3 = 13). The probability is (\frac{13}{40}).
In a board game, you roll a fair die once. What is the probability of rolling a 6 outcome?
Explanation: This question tests middle school mathematics skills, specifically calculating probability from outcomes (aligned with ISEE standards). Probability measures the likelihood of an event occurring, calculated as the ratio of favorable outcomes to total possible outcomes. In this scenario, students must recognize that a fair die has 6 faces numbered 1 through 6, with only one face showing 6 (favorable outcome). The correct answer works by calculating 1/6 or approximately 17%, showing understanding that each face has equal probability. A common distractor like 5/6 might arise from students calculating the probability of NOT rolling a 6 instead. To help students, use visual aids like dice and emphasize that "fair" means each outcome is equally likely.
A bag has 2 yellow, 3 purple, and 5 orange marbles. What is the probability of drawing purple outcome?
Explanation: This question tests middle school mathematics skills, specifically calculating probability from outcomes (aligned with ISEE standards). Probability measures the likelihood of an event occurring, calculated as the ratio of favorable outcomes to total possible outcomes. In this scenario, students must identify that there are 3 purple marbles (favorable outcomes) out of 2 + 3 + 5 = 10 total marbles. The correct answer works by calculating 3/10 or 30%, showing accurate counting and fraction formation. A common distractor like 7/10 might arise from adding the other colors instead of focusing on purple. To help students, teach them to underline or circle the specific outcome requested in the problem and double-check their counting before calculating.
In gym class, you roll a fair die once. What is the probability of an even outcome?
Explanation: This question tests middle school mathematics skills, specifically calculating probability from outcomes (aligned with ISEE standards). Probability measures the likelihood of an event occurring, calculated as the ratio of favorable outcomes to total possible outcomes. In this scenario, students must identify the even outcomes when rolling a fair die: 2, 4, and 6 (3 favorable outcomes) out of 6 total possible outcomes (1, 2, 3, 4, 5, 6). The correct answer works by calculating 3/6 = 1/2 or 50%, showing understanding that half the numbers on a die are even. A common distractor might be 1/3, assuming only two outcomes are even, or misunderstanding what constitutes an even number. To help students, teach them to list all possible outcomes systematically and identify which satisfy the given condition.
A bag has 3 red, 5 blue, and 2 green marbles. What is the probability of drawing a blue outcome?
Explanation: This question tests middle school mathematics skills, specifically calculating probability from outcomes (aligned with ISEE standards). Probability measures the likelihood of an event occurring, calculated as the ratio of favorable outcomes to total possible outcomes. In this scenario, students must identify that there are 5 blue marbles (favorable outcomes) out of 3 + 5 + 2 = 10 total marbles (total possible outcomes). The correct answer should be 5/10 = 1/2 or 50%, but the marked answer A shows 1/2 (50%) which appears correct. However, upon verification, the probability of drawing a blue marble is indeed 5/10 = 1/2, confirming answer A is correct. To help students, teach them to first count all items, then identify favorable outcomes, and simplify fractions when possible.
You spin a wheel with 8 equal sections: 3 blue, 2 red, 2 green, 1 yellow. What is the probability of blue outcome?
Explanation: This question tests middle school mathematics skills, specifically calculating probability from outcomes (aligned with ISEE standards). Probability measures the likelihood of an event occurring, calculated as the ratio of favorable outcomes to total possible outcomes. In this scenario, students must identify that there are 3 blue sections (favorable outcomes) out of 8 total equal sections on the wheel. The correct answer works by calculating 3/8 or 37.5%, accurately representing the proportion of blue sections. A common distractor like 3/5 fails by using only the number of different colors rather than counting all sections. To help students, emphasize the importance of counting all sections when they are equal in size, and practice converting fractions to percentages by multiplying by 100.
You spin a wheel with 10 equal sections: 4 prize, 6 no-prize. What is the probability of prize outcome?
Explanation: This question tests middle school mathematics skills, specifically calculating probability from outcomes (aligned with ISEE standards). Probability measures the likelihood of an event occurring, calculated as the ratio of favorable outcomes to total possible outcomes. In this scenario, students must identify that there are 4 prize sections (favorable outcomes) out of 10 total equal sections on the wheel. The correct answer works by calculating 4/10 or 40%, accurately representing the proportion of prize sections. A common distractor like 6/10 might occur if students mistakenly count the no-prize sections instead. To help students, emphasize careful reading to identify what outcome is being asked for, and practice simplifying fractions when appropriate.
You flip a fair coin three times. Outcomes are H or T each time, so 8 total. What is the probability of HHH outcome?
Explanation: This question tests middle school mathematics skills, specifically calculating probability from outcomes (aligned with ISEE standards). Probability measures the likelihood of an event occurring, calculated as the ratio of favorable outcomes to total possible outcomes. In this scenario, students must identify and count the HHH outcome from three coin flips, with 1 out of 8 possible outcomes. The correct answer works by accurately calculating the probability as 1/8, showing a clear understanding of the total outcome space. A common distractor fails by confusing with fewer flips, leading to 1/4. To help students, teach them to list all possible outcomes and practice converting between fractions and percentages. Encourage checking calculations by ensuring the probabilities sum to 1 or 100%.
A spinner has 10 equal sections: 1 prize, 9 no-prize. What is the probability of a prize outcome?
Explanation: This question tests middle school mathematics skills, specifically calculating probability from outcomes (aligned with ISEE standards). Probability measures the likelihood of an event occurring, calculated as the ratio of favorable outcomes to total possible outcomes. In this scenario, students must identify and count the prize sections on a 10-section spinner, with 1 prize out of 10. The correct answer works by accurately calculating the probability as 1/10, showing a clear understanding of the total outcome space. A common distractor fails by inverting the ratio, leading to an incorrect calculation like 9/10. To help students, teach them to list all possible outcomes and practice converting between fractions and percentages. Encourage checking calculations by ensuring the probabilities sum to 1 or 100%.
You roll a fair die for a board game. Outcomes 1–6 are each 1/6. What is the probability of rolling 6 outcome?
Explanation: This question tests middle school mathematics skills, specifically calculating probability from outcomes (aligned with ISEE standards). Probability measures the likelihood of an event occurring, calculated as the ratio of favorable outcomes to total possible outcomes. In this scenario, students must identify and count the outcome of rolling a 6 on a die, with 1 favorable out of 6. The correct answer works by accurately calculating the probability as 1/6, showing a clear understanding of the total outcome space. A common distractor fails by assuming unequal likelihood, leading to an incorrect calculation like 1/5. To help students, teach them to list all possible outcomes and practice converting between fractions and percentages. Encourage checking calculations by ensuring the probabilities sum to 1 or 100%.
A fair coin is tossed once before a quiz. Outcomes are heads or tails, each 1/2. What is the probability of not heads outcome?
Explanation: This question tests middle school mathematics skills, specifically calculating probability from outcomes (aligned with ISEE standards). Probability measures the likelihood of an event occurring, calculated as the ratio of favorable outcomes to total possible outcomes. In this scenario, students must identify and count the not heads outcome from a coin toss, which is tails, with 1 out of 2. The correct answer works by accurately calculating the probability as 1/2, showing a clear understanding of the total outcome space. A common distractor fails by assuming impossible outcomes, leading to 0%. To help students, teach them to list all possible outcomes and practice converting between fractions and percentages. Encourage checking calculations by ensuring the probabilities sum to 1 or 100%.
A fair six-sided die is rolled after school. Outcomes are 1–6, each 1/6. What is the probability of an even outcome?
Explanation: This question tests middle school mathematics skills, specifically calculating probability from outcomes (aligned with ISEE standards). Probability measures the likelihood of an event occurring, calculated as the ratio of favorable outcomes to total possible outcomes. In this scenario, students must identify and count the even numbers on a six-sided die, with 3 even outcomes out of 6. The correct answer works by accurately calculating the probability as 3/6, which simplifies to 1/2, showing a clear understanding of the total outcome space. A common distractor fails by assuming only two even numbers or miscounting, leading to an incorrect calculation like 2/6 or 1/3. To help students, teach them to list all possible outcomes and practice converting between fractions and percentages. Encourage checking calculations by ensuring the probabilities sum to 1 or 100%.
In gym, you flip a fair coin twice. Outcomes are HH, HT, TH, TT, each 1/4. What is the probability of HH outcome?
Explanation: This question tests middle school mathematics skills, specifically calculating probability from outcomes (aligned with ISEE standards). Probability measures the likelihood of an event occurring, calculated as the ratio of favorable outcomes to total possible outcomes. In this scenario, students must identify and count the outcomes from flipping a coin twice, with 1 HH out of 4 possible outcomes. The correct answer works by accurately calculating the probability as 1/4, showing a clear understanding of the total outcome space. A common distractor fails by confusing single flips with multiple, leading to an incorrect calculation like 1/2. To help students, teach them to list all possible outcomes and practice converting between fractions and percentages. Encourage checking calculations by ensuring the probabilities sum to 1 or 100%.
A fair coin is tossed once. Outcomes are heads or tails, each 1/2. What is the probability of tails outcome?
Explanation: This question tests middle school mathematics skills, specifically calculating probability from outcomes (aligned with ISEE standards). Probability measures the likelihood of an event occurring, calculated as the ratio of favorable outcomes to total possible outcomes. In this scenario, students must identify and count the tails outcome from a coin toss, with 1 tails out of 2. The correct answer works by accurately calculating the probability as 1/2, showing a clear understanding of the total outcome space. A common distractor fails by assuming more outcomes, leading to an incorrect calculation like 1/4. To help students, teach them to list all possible outcomes and practice converting between fractions and percentages. Encourage checking calculations by ensuring the probabilities sum to 1 or 100%.
Two distinct numbers are selected at random from the set {1, 2, 3, 4, 5}. What is the probability that their sum is even?
Explanation: The sum of two numbers is even if both numbers are even or both numbers are odd. The set is {1, 2, 3, 4, 5}, which has 3 odd numbers {1, 3, 5} and 2 even numbers {2, 4}. The total number of ways to choose two distinct numbers is (C(5,2) = \frac{5 \times 4}{2} = 10). Case 1: Both are odd. The number of ways to choose 2 odd numbers from 3 is (C(3,2) = 3). Case 2: Both are even. The number of ways to choose 2 even numbers from 2 is (C(2,2) = 1). The total number of favorable outcomes is (3 + 1 = 4). The probability is (\frac{4}{10} = \frac{2}{5}).
Alex has a bag with 4 red and 6 blue marbles. Beth has a bag with 3 red and 4 blue marbles. Alex draws one marble from his bag, and Beth draws one from hers. What is the probability that both marbles drawn are the same color?
Explanation: There are two ways for the marbles to be the same color: both are red, or both are blue. We calculate the probability of each case and add them. Case 1: Both red. The probability Alex draws red is (\frac{4}{10}). The probability Beth draws red is (\frac{3}{7}). The probability of both drawing red is (\frac{4}{10} \times \frac{3}{7} = \frac{12}{70}). Case 2: Both blue. The probability Alex draws blue is (\frac{6}{10}). The probability Beth draws blue is (\frac{4}{7}). The probability of both drawing blue is (\frac{6}{10} \times \frac{4}{7} = \frac{24}{70}). The total probability of drawing the same color is the sum of these probabilities: (\frac{12}{70} + \frac{24}{70} = \frac{36}{70}), which simplifies to (\frac{18}{35}).