Calculating Probability - ISEE Lower Level Quantitative Reasoning

1. A box has $6$ red, $2$ blue, and $4$ white balls. What is $P(\text{not white})$?

   $\frac{2}{3}$. This is the complement of drawing a white ball, calculated as 1 minus the probability of white.

2. What is the probability of choosing a blue marble if there are $3$ red and $7$ blue marbles?

   $\frac{7}{10}$. The probability is the ratio of blue marbles to the total marbles, assuming each marble is equally likely to be chosen.

3. A class has $18$ students: $10$ wear glasses. What is $P(\text{glasses})$ in simplest form?

   $\frac{5}{9}$. Simplify the ratio of students with glasses to total students by dividing numerator and denominator by 2.

4. If $P(E)=\frac{1}{8}$, what is $P(E)$ as a percent?

   $12.5\%$. Convert the fraction $\frac{1}{8}$ to a decimal 0.125 and then to a percent by multiplying by 100.

5. A bag has $5$ green, $4$ yellow, and $1$ purple marble. What is $P(\text{yellow})$?

   $\frac{2}{5}$. With 4 yellow out of 10 total marbles, the probability simplifies to this fraction assuming equal likelihood.

6. What is the probability formula for the complement of event $E$?

   $P(E^c)=1-P(E)$. The probability of the complement event $E^c$ is one minus the probability of $E$, as they are mutually exclusive and exhaustive.

7. A ratio of favorable to total outcomes is $9:12$. What is the probability in simplest form?

   $\frac{3}{4}$. Simplify the ratio 9:12 by dividing both parts by 3 to get the probability as a fraction.

8. A bag has $5$ green, $4$ yellow, and $1$ purple marble. What is $P(\text{not green})$?

   $\frac{1}{2}$. This is the complement of drawing a green marble, or equivalently the ratio of non-green marbles to total marbles.

9. A jar has $12$ candies: $5$ are lemon. What is $P(\text{lemon})$ in simplest form?

   $\frac{5}{12}$. The probability is the ratio of lemon candies to total candies, already in simplest form.

10. What is the probability of choosing a red marble if there are $3$ red and $7$ blue marbles?

    $\frac{3}{10}$. The probability is the ratio of red marbles to the total marbles, assuming each marble is equally likely to be chosen.

11. A deck has $52$ cards. What is the probability of drawing a heart?

    $\frac{1}{4}$. There are 13 hearts in a standard deck of 52 cards, so the probability simplifies to this fraction.

12. What is the probability of rolling a $6$ on a fair six-sided number cube?

    $\frac{1}{6}$. Each face of the die is equally likely, so the probability is one favorable outcome divided by six total outcomes.

13. What is the probability of rolling a number greater than $4$ on a fair six-sided number cube?

    $\frac{1}{3}$. Two numbers greater than 4 (5 and 6) out of six possible outcomes on the die yield this probability.

14. A spinner has $8$ equal sections, and $3$ are shaded. What is $P(\text{shaded})$?

    $\frac{3}{8}$. Each section is equally likely, so the probability is the number of shaded sections divided by total sections.

15. What is the probability of flipping heads twice in $2$ fair coin flips?

    $\frac{1}{4}$. The probability of heads on each independent flip is $\frac{1}{2}$, so for two heads it is $\left(\frac{1}{2}\right)^2$.

16. If $P(E)=\frac{3}{4}$, what is $P(E)$ as a decimal?

    $0.75$. Divide the numerator 3 by the denominator 4 to express the probability as a decimal.

17. A drawer has $9$ socks: $4$ black and $5$ white. What is $P(\text{black})$?

    $\frac{4}{9}$. The probability is the ratio of black socks to total socks, assuming each sock is equally likely to be drawn.

18. What is the probability of rolling an even number on a fair six-sided number cube?

    $\frac{1}{2}$. Three even numbers (2, 4, 6) out of six possible outcomes on the die yield this probability.

19. What is the probability of flipping heads on a fair coin?

    $\frac{1}{2}$. A fair coin has two equally likely outcomes, heads and tails, so the probability of heads is one out of two.

20. If $P(E)=\frac{2}{5}$, what is $P(E^c)$?

    $\frac{3}{5}$. The complement probability is found by subtracting $P(E)$ from 1.

21. A box has $6$ red, $2$ blue, and $4$ white balls. What is $P(\text{red or blue})$?

    $\frac{2}{3}$. Add red and blue balls for favorable outcomes, then divide by total balls to get the probability.

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