Comparing Probabilities - ISEE Middle Level Mathematics Achievement

1. Which probability is larger: $\frac{9}{25}$ or $35\%$?

   $\frac{9}{25}$. Converting shows $\frac{9}{25}=0.36$ exceeds $35\%=0.35$.

2. Which probability is larger: $P(E)$ or $P(E^c)$ if $P(E)=0.6$?

   $P(E)$. With $P(E)=0.6$, it exceeds $P(E^c)=0.4$.

3. Which is more likely: rolling an even number or rolling a number greater than $4$ on a fair die?

   Rolling an even number. Even numbers (2,4,6) have probability $\frac{3}{6}=0.5$, exceeding numbers greater than 4 (5,6) at $\frac{2}{6} \approx 0.333$.

4. Which probability is larger: $\frac{13}{50}$ or $0.27$?

   $0.27$. $0.27$ is greater than $\frac{13}{50}=0.26$ in decimal form.

5. Which probability is larger: $0.37$ or $\frac{3}{10}$?

   $0.37$. $0.37$ exceeds $\frac{3}{10}=0.3$ when compared directly as decimals.

6. Which probability is larger: $\frac{11}{30}$ or $\frac{3}{8}$?

   $\frac{3}{8}$. Converting to decimals indicates $\frac{3}{8}=0.375$ is greater than $\frac{11}{30} \approx 0.367$.

7. What is the probability of an event $E$ in terms of favorable and total outcomes?

   $P(E)=\frac{\text{favorable}}{\text{total}}$. Probability measures the likelihood as the ratio of favorable outcomes to total possible outcomes in a uniform sample space.

8. Which probability is larger: $\frac{4}{9}$ or $0.45$?

   $0.45$. $0.45$ slightly exceeds $\frac{4}{9} \approx 0.444$ when compared as decimals.

9. Which probability is larger: $\frac{1}{4}$ or $0.26$?

   $0.26$. $0.26$ exceeds $\frac{1}{4}=0.25$ when compared directly.

10. Which probability is larger: $P(E)$ or $P(E^c)$ if $P(E)=\frac{2}{5}$?

    $P(E^c)$. Given $P(E)=\frac{2}{5}=0.4$, $P(E^c)=0.6$ is larger.

11. Which probability is larger: $\frac{3}{8}$ or $\frac{2}{5}$?

    $\frac{2}{5}$. Converting to decimals shows $\frac{2}{5}=0.4$ exceeds $\frac{3}{8}=0.375$.

12. Which probability is closest to $\frac{1}{2}$: $0.49$, $0.52$, or $0.6$?

    $0.49$. $0.49$ is closest to $0.5$, being only $0.01$ away compared to $0.02$ for $0.52$ and $0.1$ for $0.6$.

13. Which probability is larger: $\frac{5}{8}$ or $62\%$?

    $\frac{5}{8}$. Converting shows $\frac{5}{8}=0.625$ exceeds $62\%=0.62$.

14. Which probability is larger: $\frac{3}{7}$ or $\frac{4}{9}$?

    $\frac{4}{9}$. Converting to decimals reveals $\frac{4}{9} \approx 0.444$ is greater than $\frac{3}{7} \approx 0.429$.

15. Which probability is larger: $\frac{7}{20}$ or $0.36$?

    $0.36$. $0.36$ is greater than $\frac{7}{20}=0.35$ when compared as decimals.

16. What is the complement rule for an event $E$?

    $P(E^c)=1-P(E)$. The complement rule states that the probability of an event not occurring is one minus the probability of it occurring.

17. Which probability is larger: $\frac{17}{40}$ or $0.41$?

    $\frac{17}{40}$. Converting shows $\frac{17}{40}=0.425$ exceeds $0.41$.

18. Which probability is larger: $\frac{5}{12}$ or $\frac{2}{3}$?

    $\frac{2}{3}$. Converting to decimals reveals $\frac{2}{3} \approx 0.667$ is greater than $\frac{5}{12} \approx 0.417$.

19. Which probability is larger: $0.125$ or $\frac{1}{9}$?

    $0.125$. $0.125$ exceeds $\frac{1}{9} \approx 0.111$ when compared as decimals.

20. Which probability is larger: $45\%$ or $0.43$?

    $45\%$. Converting $45\%$ to $0.45$ shows it exceeds $0.43$.

21. Which is more likely: $3$ heads in $3$ fair coin flips or $0$ heads in $3$ flips?

    They are equally likely. Both outcomes have probability $\frac{1}{8}$ in a binomial distribution with fair coins.

22. Which probability is closest to $1$: $\frac{7}{8}$, $\frac{5}{6}$, or $\frac{9}{10}$?

    $\frac{9}{10}$. $\frac{9}{10}=0.9$ is nearest to $1$, compared to $\frac{7}{8}=0.875$ and $\frac{5}{6} \approx 0.833$.

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