Comparing Probabilities
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ISEE Middle Level: Mathematics Achievement › Comparing Probabilities
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 probabilities of Event X and Event Y are equal.
The probability of Event Y is greater.
The sum of the probabilities is exactly \(\frac{1}{2}\).
The probability of Event X is greater.
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) = \(\frac{1}{6}\). 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) = \(\frac{1}{6} + \frac{1}{6} = \frac{2}{6} = \frac{1}{3}\). Comparing the probabilities, \(\frac{1}{3} > \frac{1}{6}\), so the probability of Event Y is greater.
Game: $P(\text{spin a 1})=20%$ and $P(\text{spin a 2})=\frac{1}{4}$. Which event is more likely?
Spin a 1, 20%
Spin a 1, $\frac{1}{4}$
Spin a 2, $\frac{1}{4}$
They are equally likely events
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.
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.
The probability of Event M is greater.
The probability of Event M is half the probability of Event N.
The probability of Event N is greater.
The probabilities of Event M and Event N are equal.
Explanation
There are \(6 \times 6 = 36\) 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) = \(\frac{6}{36} = \frac{1}{6}\). 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) = \(\frac{6}{36} = \frac{1}{6}\). The probabilities are equal.
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?
The probabilities are equal for both classes.
Mr. Smith's class, because the ratio of boys to girls is higher.
Ms. Jones's class, because the probability fraction is larger.
Ms. Jones's class, because it has more boys in total.
Explanation
The probability of choosing a boy from Mr. Smith's class is \(\frac{10}{10+15} = \frac{10}{25} = \frac{2}{5}\). The probability of choosing a boy from Ms. Jones's class is \(\frac{12}{12+16} = \frac{12}{28} = \frac{3}{7}\). To compare \(\frac{2}{5}\) and \(\frac{3}{7}\), find a common denominator, which is 35. \(\frac{2}{5} = \frac{14}{35}\) and \(\frac{3}{7} = \frac{15}{35}\). Since \(\frac{15}{35} > \frac{14}{35}\), there is a greater probability of choosing a boy from Ms. Jones's class.
Bag 1: $P(\text{black marble})=\frac{3}{10}$. Bag 2: $P(\text{white marble})=\frac{1}{4}$. Which is more likely?
Black marble, $\frac{3}{10}$
White marble, $\frac{3}{10}$
White marble, $\frac{1}{4}$
Both events are equally 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 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.
Weather app: $P(\text{cloudy})=70%$ and $P(\text{sunny})=25%$. Compare the likelihood of these events.
Cloudy and sunny are equally likely
Sunny, 70% is more likely
Cloudy, 70% is more likely
Sunny, 25% 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.
Game: $P(\text{win a small prize})=15%$ and $P(\text{win a big prize})=5%$. Which is more likely?
Win a big prize, 5%
Win a small prize, 15%
Both prizes are equally likely
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.
In a game, $P(\text{green spinner})=25%$ and $P(\text{yellow spinner})=30%$. Which is more likely?
Both colors are equally likely
Green is more likely because it’s brighter
Landing on green, 25%
Landing on yellow, 30%
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.
In a game, $P(\text{red die}=6)=\frac{1}{6}$ and $P(\text{blue die}=5)=\frac{1}{3}$. Which is more likely?
Blue die shows 5, $\frac{1}{3}$
Both events are equally likely
Red die shows 6, $\frac{1}{6}$
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.
Weather: $P(\text{rain})=35%$ and $P(\text{thunderstorms})=20%$. Based on probabilities, which has a greater chance?
They are equally likely today
Thunderstorms, 35%
Thunderstorms, 20%
Rain, 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.