ISEE Middle Level Quiz: Comparing Probabilities
20 questions · exam conditions
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Comparing ProbabilitiesQuestion 1 of 20

A fair six-sided die is rolled once. Let Event X be the event that the number rolled is both an even number and a prime number. Let Event Y be the event that the number rolled is a 5 or a 6. Compare the probability of Event X and Event Y.

The probability of Event X is greater.
The probability of Event Y is greater.
The probabilities of Event X and Event Y are equal.
The sum of the probabilities is exactly 12\frac{1}{2}.
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ISEE Middle Level Quiz

ISEE Middle Level Quiz: Comparing Probabilities

Practice Comparing Probabilities in ISEE Middle Level with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Comparing Probabilities, giving you a quick way to practice the rules, question types, and explanations that matter most for ISEE Middle Level.

How to use this quiz

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.

All questions

Question 1

A fair six-sided die is rolled once. Let Event X be the event that the number rolled is both an even number and a prime number. Let Event Y be the event that the number rolled is a 5 or a 6. Compare the probability of Event X and Event Y.

  1. The probability of Event X is greater.
  2. The probability of Event Y is greater. (correct answer)
  3. The probabilities of Event X and Event Y are equal.
  4. The sum of the probabilities is exactly 12\frac{1}{2}.
Explanation: For Event X, we need a number that is both even and prime. The only even prime number is 2. So, there is only one favorable outcome for Event X. The probability is P(X) = 16\frac{1}{6}. For Event Y, the number rolled is a 5 or a 6. These are two distinct outcomes. The probability is P(Y) = P(5) + P(6) = 16+16=26=13\frac{1}{6} + \frac{1}{6} = \frac{2}{6} = \frac{1}{3}. Comparing the probabilities, 13>16\frac{1}{3} > \frac{1}{6}, so the probability of Event Y is greater.

Question 2

Game: P(spin a 1)=20%P(\text{spin a 1})=20\% and P(spin a 2)=14P(\text{spin a 2})=\frac{1}{4}. Which event is more likely?

  1. Spin a 1, 20%
  2. Spin a 2, 14\frac{1}{4} (correct answer)
  3. They are equally likely events
  4. Spin a 1, 14\frac{1}{4}
Explanation: This question tests middle school mathematics skills in comparing probabilities. Probability comparison involves determining which event is more likely by comparing numerical values representing likelihood. In the provided scenario, students are given probabilities of spinning a 1 (20%) and a 2 (1/4), and must decide which event is more likely using the given numbers. The correct choice clearly identifies spinning a 2 with the higher probability of 1/4, demonstrating understanding of basic probability concepts. A common mistake is choosing the wrong event due to misunderstanding of fractions or percentages, like choosing a smaller number as larger. Teaching strategies include practicing probability with real-life contexts, using visual aids like fraction bars or pie charts to compare probabilities, and reinforcing the concept that higher numbers signify greater likelihood.

Question 3

Two fair six-sided dice are rolled. Let Event M be the event that the sum of the numbers on the two dice is 7. Let Event N be the event that the number on the first die is a 3. Compare the probabilities of Event M and Event N.

  1. The probability of Event M is greater.
  2. The probability of Event N is greater.
  3. The probabilities of Event M and Event N are equal. (correct answer)
  4. The probability of Event M is half the probability of Event N.
Explanation: There are 6×6=366 \times 6 = 36 possible outcomes when rolling two dice. For Event M, the combinations that sum to 7 are (1,6), (2,5), (3,4), (4,3), (5,2), and (6,1). There are 6 favorable outcomes, so P(M) = 636=16\frac{6}{36} = \frac{1}{6}. For Event N, the first die must be a 3. The second die can be any of the 6 numbers. The combinations are (3,1), (3,2), (3,3), (3,4), (3,5), and (3,6). There are 6 favorable outcomes, so P(N) = 636=16\frac{6}{36} = \frac{1}{6}. The probabilities are equal.

Question 4

Mr. Smith's class has 10 boys and 15 girls. Ms. Jones's class has 12 boys and 16 girls. A student is chosen at random from each class. From which class is there a greater probability of choosing a boy?

  1. Mr. Smith's class, because the ratio of boys to girls is higher.
  2. Ms. Jones's class, because the probability fraction is larger. (correct answer)
  3. The probabilities are equal for both classes.
  4. Ms. Jones's class, because it has more boys in total.
Explanation: The probability of choosing a boy from Mr. Smith's class is 1010+15=1025=25\frac{10}{10+15} = \frac{10}{25} = \frac{2}{5}. The probability of choosing a boy from Ms. Jones's class is 1212+16=1228=37\frac{12}{12+16} = \frac{12}{28} = \frac{3}{7}. To compare 25\frac{2}{5} and 37\frac{3}{7}, find a common denominator, which is 35. 25=1435\frac{2}{5} = \frac{14}{35} and 37=1535\frac{3}{7} = \frac{15}{35}. Since 1535>1435\frac{15}{35} > \frac{14}{35}, there is a greater probability of choosing a boy from Ms. Jones's class.

Question 5

Team A has P(win)=65%P(\text{win})=65\% and Team B has P(win)=60%P(\text{win})=60\%. Which is more likely?

  1. Team B wins, 60%
  2. Both teams are equally likely to win
  3. Team A wins, 65% (correct answer)
  4. Neither team can win the match
Explanation: This question tests middle school mathematics skills in comparing probabilities. Probability comparison involves determining which event is more likely by comparing numerical values representing likelihood. In the provided scenario, students are given probabilities of Team A winning (65%) and Team B winning (60%), and must decide which is more probable using the given numbers. The correct choice clearly identifies Team A winning with the higher probability of 65%, demonstrating understanding of basic probability concepts. A common mistake is choosing the wrong event due to misunderstanding of percentages, like choosing a smaller number as larger. Teaching strategies include practicing probability with real-life contexts, using visual aids like fraction bars or pie charts to compare probabilities, and reinforcing the concept that higher numbers signify greater likelihood.

Question 6

A spinner has 4 equal sections colored red, blue, green, and yellow. The theoretical probability of landing on red is 14\frac{1}{4}. In an experiment, the spinner was spun 50 times and landed on red 15 times. Let P(T) be the theoretical probability of landing on red and P(E) be the experimental probability from this experiment. How do P(T) and P(E) compare?

  1. P(T) is greater than P(E).
  2. They cannot be compared because one is theoretical.
  3. P(T) and P(E) are equal.
  4. P(E) is greater than P(T). (correct answer)
Explanation: When you encounter probability comparison questions, you need to distinguish between theoretical probability (what should happen mathematically) and experimental probability (what actually happened in trials). The theoretical probability P(T) is given as 14\frac{1}{4} or 0.25, since the spinner has 4 equal sections and red occupies one of them. To find the experimental probability P(E), you divide the actual occurrences by the total trials: P(E)=1550=310=0.30P(E) = \frac{15}{50} = \frac{3}{10} = 0.30 Comparing these values: P(E) = 0.30 and P(T) = 0.25, so P(E) is greater than P(T). Looking at the wrong answers: Choice A incorrectly reverses the relationship—while 0.25 < 0.30, not the other way around. Choice B reflects a fundamental misunderstanding; theoretical and experimental probabilities are absolutely comparable since both are expressed as fractions or decimals between 0 and 1. That's exactly what probability questions test! Choice C suggests the values are equal, but 0.30 ≠ 0.25. Choice D correctly identifies that the experimental probability (0.30) exceeds the theoretical probability (0.25). Study tip: Remember that experimental probability rarely equals theoretical probability, especially with smaller sample sizes. The experimental results fluctuate around the theoretical value—sometimes higher, sometimes lower. Always calculate both probabilities as decimals to make comparison easier, and don't assume they can't be compared just because one comes from theory and one from experiment.

Question 7

In a cafeteria, Line A has 30 students, 12 of whom are buying pizza. Line B has 25 students, 10 of whom are buying pizza. Compare the probability that a randomly selected student from Line A is NOT buying pizza to the probability that a randomly selected student from Line B IS buying pizza.

  1. The probability of the student from Line A not buying pizza is greater. (correct answer)
  2. The probability of the student from Line B buying pizza is greater.
  3. The two probabilities are equal.
  4. The relationship cannot be determined with the given numbers.
Explanation: In Line A, 12 out of 30 students are buying pizza, so 30−12=1830 - 12 = 18 students are not buying pizza. The probability is 1830=35\frac{18}{30} = \frac{3}{5}. In Line B, 10 out of 25 students are buying pizza. The probability is 1025=25\frac{10}{25} = \frac{2}{5}. Comparing the two probabilities, 35>25\frac{3}{5} > \frac{2}{5}. Therefore, the probability of the student from Line A not buying pizza is greater.

Question 8

A jar contains 5 red candies and 5 green candies. Let P1(Red) be the probability of picking a red candy on the first draw. A green candy is then drawn from the jar and is not replaced. Let P2(Red) be the probability of picking a red candy on the second draw. Compare P1(Red) and P2(Red).

  1. P1(Red) is greater than P2(Red).
  2. The relationship depends on which green candy was picked.
  3. P1(Red) is equal to P2(Red).
  4. P2(Red) is greater than P1(Red). (correct answer)
Explanation: This question tests your understanding of conditional probability and how removing items from a sample affects future probabilities. Let's calculate each probability step by step. Initially, the jar contains 10 candies total: 5 red and 5 green. So P1(Red)=510=12=0.5P_1(\text{Red}) = \frac{5}{10} = \frac{1}{2} = 0.5. After a green candy is drawn and not replaced, the jar now contains 9 candies total: 5 red and 4 green. Therefore, P2(Red)=59≈0.56P_2(\text{Red}) = \frac{5}{9} \approx 0.56. Since 59>12\frac{5}{9} > \frac{1}{2}, we have P2(Red)>P1(Red)P_2(\text{Red}) > P_1(\text{Red}). Looking at the wrong answers: Choice A claims P1(Red)P_1(\text{Red}) is greater, but we've shown the opposite is true. Choice B suggests the relationship depends on which specific green candy was picked, but this is incorrect—removing any green candy has the same effect on the probability calculation. Choice C states the probabilities are equal, but 12≠59\frac{1}{2} \neq \frac{5}{9}. The key insight is that removing a green candy increases the proportion of red candies remaining in the jar. When you remove an item that's not the type you're interested in, you increase the probability of selecting your desired type on subsequent draws. Study tip: In probability problems involving drawing without replacement, always ask yourself: "Does removing this item help or hurt my chances of getting what I want next time?" Removing items you don't want increases your odds; removing items you do want decreases them.

Question 9

You have two bags. Bag 1 contains 3 red balls and 2 blue balls. Bag 2 contains 2 red balls and 3 blue balls. Compare the probability of drawing a single red ball from Bag 1 with the probability of drawing one red ball from Bag 1 and then one red ball from Bag 2.

  1. The two probabilities are equal.
  2. The probability of drawing two red balls is greater.
  3. The probability of drawing a single red ball from Bag 1 is greater. (correct answer)
  4. The probabilities are complements of each other.
Explanation: When comparing probabilities, you need to calculate each scenario separately and then determine which value is larger. This question tests your understanding of single events versus compound events. For the first scenario, you're finding the probability of drawing one red ball from Bag 1. Bag 1 has 3 red balls and 2 blue balls (5 total), so P(red from Bag 1) = 35=0.6\frac{3}{5} = 0.6. For the second scenario, you need one red ball from Bag 1 AND one red ball from Bag 2. Since these are independent events, you multiply the probabilities. Bag 2 has 2 red balls and 3 blue balls (5 total), so P(red from Bag 2) = 25=0.4\frac{2}{5} = 0.4. Therefore, P(red from Bag 1 AND red from Bag 2) = 35×25=625=0.24\frac{3}{5} \times \frac{2}{5} = \frac{6}{25} = 0.24. Comparing the results: 0.6 > 0.24, so drawing a single red ball from Bag 1 has the greater probability. Choice A is wrong because 35≠625\frac{3}{5} \neq \frac{6}{25}. Choice B is incorrect because the compound probability (0.24) is actually smaller than the single event probability (0.6). Choice D is wrong because complementary probabilities sum to 1, but 0.6+0.24=0.84≠10.6 + 0.24 = 0.84 \neq 1. Remember: when you multiply probabilities for independent events, the result is always smaller than either individual probability. Compound events are generally less likely than single events.

Question 10

Three separate events are considered. Event A: A fair coin is flipped twice, and at least one of the flips is tails. Event B: A card is drawn from a standard 52-card deck, and it is a red card. Event C: A fair six-sided die is rolled, and the result is a number greater than 2. Which statement correctly compares the probabilities?

  1. P(A) > P(C) > P(B) (correct answer)
  2. P(A) > P(B) > P(C)
  3. P(C) > P(A) > P(B)
  4. P(A) = P(C) > P(B)
Explanation: Calculate each probability. P(A): The only outcome without at least one tail is getting two heads (HH). P(HH) = 12×12=14\frac{1}{2} \times \frac{1}{2} = \frac{1}{4}. So, P(A) = 1−14=341 - \frac{1}{4} = \frac{3}{4}. P(B): Half the deck is red cards (26 out of 52), so P(B) = 2652=12\frac{26}{52} = \frac{1}{2}. P(C): Numbers greater than 2 are 3, 4, 5, 6. There are 4 such outcomes, so P(C) = 46=23\frac{4}{6} = \frac{2}{3}. Now compare the values: P(A) = 0.75, P(B) = 0.5, P(C) ≈ 0.667. The correct order is P(A) > P(C) > P(B).

Question 11

Bag 1: P(black marble)=310P(\text{black marble})=\frac{3}{10}. Bag 2: P(white marble)=14P(\text{white marble})=\frac{1}{4}. Which is more likely?

  1. White marble, 14\frac{1}{4}
  2. Both events are equally likely
  3. Black marble, 310\frac{3}{10} (correct answer)
  4. White marble, 310\frac{3}{10}
Explanation: This question tests middle school mathematics skills in comparing probabilities. Probability comparison involves determining which event is more likely by comparing numerical values representing likelihood. In the provided scenario, students are given probabilities of drawing a black marble (3/10) and a white marble (1/4), and must decide which is more probable using the given numbers. The correct choice clearly identifies the black marble with the higher probability of 3/10, demonstrating understanding of basic probability concepts. A common mistake is choosing the wrong event due to misunderstanding of fractions, like choosing a smaller number as larger. Teaching strategies include practicing probability with real-life contexts, using visual aids like fraction bars or pie charts to compare probabilities, and reinforcing the concept that higher numbers signify greater likelihood.

Question 12

Weather app: P(cloudy)=70%P(\text{cloudy})=70\% and P(sunny)=25%P(\text{sunny})=25\%. Compare the likelihood of these events.

  1. Sunny, 25% is more likely
  2. Cloudy, 70% is more likely (correct answer)
  3. Cloudy and sunny are equally likely
  4. Sunny, 70% is more likely
Explanation: This question tests middle school mathematics skills in comparing probabilities. Probability comparison involves determining which event is more likely by comparing numerical values representing likelihood. In the provided scenario, students are given probabilities of cloudy (70%) and sunny (25%), and must compare the likelihood using the given numbers. The correct choice clearly identifies cloudy with the higher probability of 70%, demonstrating understanding of basic probability concepts. A common mistake is choosing the wrong event due to misunderstanding of percentages, like choosing a smaller number as larger. Teaching strategies include practicing probability with real-life contexts, using visual aids like fraction bars or pie charts to compare probabilities, and reinforcing the concept that higher numbers signify greater likelihood.

Question 13

Game: P(win a small prize)=15%P(\text{win a small prize})=15\% and P(win a big prize)=5%P(\text{win a big prize})=5\%. Which is more likely?

  1. Win a big prize, 5%
  2. Win a small prize, 15% (correct answer)
  3. Both prizes are equally likely
  4. Big prize is more likely because it's bigger
Explanation: This question tests middle school mathematics skills in comparing probabilities. Probability comparison involves determining which event is more likely by comparing numerical values representing likelihood. In the provided scenario, students are given probabilities of winning a small prize (15%) and a big prize (5%), and must decide which is more probable using the given numbers. The correct choice clearly identifies winning a small prize with the higher probability of 15%, demonstrating understanding of basic probability concepts. A common mistake is choosing the wrong event due to misunderstanding of percentages, like choosing a smaller number as larger. Teaching strategies include practicing probability with real-life contexts, using visual aids like fraction bars or pie charts to compare probabilities, and reinforcing the concept that higher numbers signify greater likelihood.

Question 14

In a game, P(green spinner)=25%P(\text{green spinner})=25\% and P(yellow spinner)=30%P(\text{yellow spinner})=30\%. Which is more likely?

  1. Landing on green, 25%
  2. Landing on yellow, 30% (correct answer)
  3. Both colors are equally likely
  4. Green is more likely because it's brighter
Explanation: This question tests middle school mathematics skills in comparing probabilities. Probability comparison involves determining which event is more likely by comparing numerical values representing likelihood. In the provided scenario, students are given probabilities of landing on green (25%) and yellow (30%) on a spinner, and must decide which is more probable using the given numbers. The correct choice clearly identifies landing on yellow with the higher probability of 30%, demonstrating understanding of basic probability concepts. A common mistake is choosing the wrong event due to misunderstanding of percentages, like choosing a smaller number as larger. Teaching strategies include practicing probability with real-life contexts, using visual aids like fraction bars or pie charts to compare probabilities, and reinforcing the concept that higher numbers signify greater likelihood.

Question 15

In a game, P(red die=6)=16P(\text{red die}=6)=\frac{1}{6} and P(blue die=5)=13P(\text{blue die}=5)=\frac{1}{3}. Which is more likely?

  1. Red die shows 6, 16\frac{1}{6}
  2. Both events are equally likely
  3. Blue die shows 5, 13\frac{1}{3} (correct answer)
  4. Neither event can happen at all
Explanation: This question tests middle school mathematics skills in comparing probabilities. Probability comparison involves determining which event is more likely by comparing numerical values representing likelihood. In the provided scenario, students are given probabilities of rolling a 6 on a red die (1/6) and a 5 on a blue die (1/3), and must decide which is more probable using the given numbers. The correct choice clearly identifies the blue die showing 5 with the higher probability of 1/3, demonstrating understanding of basic probability concepts. A common mistake is choosing the wrong event due to misunderstanding of fractions, like assuming 1/6 is larger than 1/3. Teaching strategies include practicing probability with real-life contexts, using visual aids like fraction bars or pie charts to compare probabilities, and reinforcing the concept that higher numbers signify greater likelihood.

Question 16

Weather: P(rain)=35%P(\text{rain})=35\% and P(thunderstorms)=20%P(\text{thunderstorms})=20\%. Based on probabilities, which has a greater chance?

  1. Thunderstorms, 20%
  2. Rain, 35% (correct answer)
  3. They are equally likely today
  4. Thunderstorms, 35%
Explanation: This question tests middle school mathematics skills in comparing probabilities. Probability comparison involves determining which event is more likely by comparing numerical values representing likelihood. In the provided scenario, students are given probabilities of rain (35%) and thunderstorms (20%), and must decide which has a greater chance using the given numbers. The correct choice clearly identifies rain with the higher probability of 35%, demonstrating understanding of basic probability concepts. A common mistake is choosing the wrong event due to misunderstanding of percentages, like choosing a smaller number as larger. Teaching strategies include practicing probability with real-life contexts, using visual aids like fraction bars or pie charts to compare probabilities, and reinforcing the concept that higher numbers signify greater likelihood.

Question 17

Two bags: P(Bag A red)=512P(\text{Bag A red})=\frac{5}{12} and P(Bag B red)=13P(\text{Bag B red})=\frac{1}{3}. Which is more likely?

  1. Bag B red, 13\frac{1}{3}
  2. Bag A red, 512\frac{5}{12} (correct answer)
  3. Both bags have equal red chance
  4. Bag A red, 13\frac{1}{3}
Explanation: This question tests middle school mathematics skills in comparing probabilities. Probability comparison involves determining which event is more likely by comparing numerical values representing likelihood. In the provided scenario, students are given probabilities of drawing red from Bag A (5/12) and Bag B (1/3), and must decide which is more probable using the given numbers. The correct choice clearly identifies Bag A red with the higher probability of 5/12, demonstrating understanding of basic probability concepts. A common mistake is choosing the wrong event due to misunderstanding of fractions, like assuming 1/3 is larger than 5/12. Teaching strategies include practicing probability with real-life contexts, using visual aids like fraction bars or pie charts to compare probabilities, and reinforcing the concept that higher numbers signify greater likelihood.

Question 18

Dice: P(blue die=4)=16P(\text{blue die}=4)=\frac{1}{6} and P(red die is even)=12P(\text{red die is even})=\frac{1}{2}. Which is more likely?

  1. Blue die shows 4, 16\frac{1}{6}
  2. Both events are equally likely
  3. Red die is even, 12\frac{1}{2} (correct answer)
  4. Blue die shows 4, 12\frac{1}{2}
Explanation: This question tests middle school mathematics skills in comparing probabilities. Probability comparison involves determining which event is more likely by comparing numerical values representing likelihood. In the provided scenario, students are given probabilities of a blue die showing 4 (1/6) and a red die being even (1/2), and must decide which is more probable using the given numbers. The correct choice clearly identifies the red die being even with the higher probability of 1/2, demonstrating understanding of basic probability concepts. A common mistake is choosing the wrong event due to misunderstanding of fractions, like choosing a smaller number as larger. Teaching strategies include practicing probability with real-life contexts, using visual aids like fraction bars or pie charts to compare probabilities, and reinforcing the concept that higher numbers signify greater likelihood.

Question 19

Sports: P(Tigers win)=710P(\text{Tigers win})=\frac{7}{10} and P(Lions win)=60%P(\text{Lions win})=60\%. Which is more likely?

  1. Lions win, 60%
  2. Tigers win, 710\frac{7}{10} (correct answer)
  3. Both wins are equally likely
  4. Lions win, 710\frac{7}{10}
Explanation: This question tests middle school mathematics skills in comparing probabilities. Probability comparison involves determining which event is more likely by comparing numerical values representing likelihood. In the provided scenario, students are given probabilities of Tigers winning (7/10) and Lions winning (60%), and must decide which is more probable using the given numbers. The correct choice clearly identifies Tigers winning with the higher probability of 7/10, demonstrating understanding of basic probability concepts. A common mistake is choosing the wrong event due to misunderstanding of fractions or percentages, like choosing a smaller number as larger. Teaching strategies include practicing probability with real-life contexts, using visual aids like fraction bars or pie charts to compare probabilities, and reinforcing the concept that higher numbers signify greater likelihood.

Question 20

A bag has P(red marble)=25P(\text{red marble})=\frac{2}{5} and another has P(blue marble)=12P(\text{blue marble})=\frac{1}{2}. Which is more likely?

  1. Red marble, 25\frac{2}{5}
  2. Blue marble, 12\frac{1}{2} (correct answer)
  3. Both draws are equally likely
  4. Red marble, 12\frac{1}{2}
Explanation: This question tests middle school mathematics skills in comparing probabilities. Probability comparison involves determining which event is more likely by comparing numerical values representing likelihood. In the provided scenario, students are given probabilities of drawing a red marble (2/5) and a blue marble (1/2), and must decide which is more probable using the given numbers. The correct choice clearly identifies the blue marble with the higher probability of 1/2, demonstrating understanding of basic probability concepts. A common mistake is choosing the wrong event due to misunderstanding of fractions, like assuming 2/5 is larger than 1/2. Teaching strategies include practicing probability with real-life contexts, using visual aids like fraction bars or pie charts to compare probabilities, and reinforcing the concept that higher numbers signify greater likelihood.