Medium ACT Probability Practice Questions

Mastering Medium ACT Probability Practice Questions is essential for students aiming for a score in the 25–30 range on the math section. Probability on the ACT measures your ability to calculate the likelihood of specific outcomes using ratios, counting principles, and logical reasoning. While basic probability involves simple fractions, medium-level questions often introduce independent and dependent events, the "complement" rule, and geometric probability. Preparing with high-quality ACT Prep resources can help you identify these patterns quickly during the exam.

Concept Explanation

Probability is the numerical representation of the likelihood that a specific event will occur, calculated as the ratio of desired outcomes to the total number of possible outcomes. The fundamental formula for probability is $P(E) = \frac{ \text{Number of favorable outcomes}}{ \text{Total number of possible outcomes}}$. All probabilities range from 0 (impossible) to 1 (certain). When solving medium-level problems, you must distinguish between independent events, where the outcome of one does not affect the other, and dependent events, where the total sample space changes. For independent events, you multiply the individual probabilities: $P(A \text{ and } B) = P(A) \times P(B)$. Another critical tool is the complement rule, which states that the probability of an event not occurring is $1 - P(E)$. This is particularly useful when a question asks for the probability of something happening "at least once." For a deeper dive into how these concepts overlap with other data topics, you may find ACT Statistics Practice Questions with Answers helpful for your study sessions.

Solved Examples

1. Example 1: Independent Events
   A bag contains 4 red marbles, 3 blue marbles, and 5 green marbles. If two marbles are drawn with replacement, what is the probability that both are blue?
   
   1. Calculate the total number of marbles: $4 + 3 + 5 = 12$.
   2. Find the probability of drawing one blue marble: $P( \text{blue}) = \frac{3}{12} = \frac{1}{4}$.
   3. Since the marble is replaced, the probability remains the same for the second draw.
   4. Multiply the probabilities: $\frac{1}{4} \times \frac{1}{4} = \frac{1}{16}$.
   5. The final answer is $\frac{1}{16}$.
2. Example 2: Dependent Events
   A committee of 2 people is to be chosen from a group of 5 men and 3 women. What is the probability that both people chosen are women?
   
   1. Total number of people is $5 + 3 = 8$.
   2. Probability the first person is a woman: $\frac{3}{8}$.
   3. After one woman is chosen, there are 2 women left and 7 total people remaining.
   4. Probability the second person is a woman: $\frac{2}{7}$.
   5. Multiply the probabilities: $\frac{3}{8} \times \frac{2}{7} = \frac{6}{56}$.
   6. Simplify the fraction: $\frac{3}{28}$.
3. Example 3: Geometric Probability
   A square target has a side length of 10 inches. Inside the square is a circle with a radius of 3 inches. If a dart hits the target at random, what is the probability it lands inside the circle?
   
   1. Calculate the area of the square (total area): $10^2 = 100$ sq in.
   2. Calculate the area of the circle (target area): $A = \pi r^2 = \pi(3)^2 = 9\pi$.
   3. Set up the ratio: $\frac{9\pi}{100}$.
   4. The probability is $\frac{9\pi}{100}$, which is approximately 0.283.

Practice Questions

1. A box contains 6 black pens, 4 blue pens, and 5 red pens. If one pen is chosen at random, what is the probability that it is NOT black?
2. Two fair six-sided dice are rolled. What is the probability that the sum of the numbers shown on the top faces is exactly 9?
3. In a class of 30 students, 12 are in the band, 15 are in the choir, and 5 are in both. If a student is picked at random, what is the probability they are in neither the band nor the choir?

4. A spinner is divided into 8 equal sections numbered 1 through 8. What is the probability that the spinner lands on a prime number?
5. A jar contains 10 gold coins and 20 silver coins. If two coins are picked at random without replacement, what is the probability that both are gold?
6. A rectangular garden measures 20 feet by 30 feet. A small circular fountain with a diameter of 4 feet is located in the center. If a bird lands randomly in the garden, what is the probability it lands in the fountain?
7. In a group of 50 people, 20 speak Spanish, 25 speak French, and 10 speak both. What is the probability that a person chosen at random speaks Spanish but NOT French?
8. If the probability of rain tomorrow is $0.35$, what is the probability that it does NOT rain tomorrow?
9. A bag contains 5 red, 3 green, and 2 blue marbles. If three marbles are drawn with replacement, what is the probability of drawing red, then green, then blue in that specific order?
10. A drawer contains 8 loose white socks and 6 loose black socks. What is the probability of pulling out a matching pair of white socks if two socks are drawn without replacement?

Answers & Explanations

1. Answer: $\frac{3}{5}$
   Total pens = $6 + 4 + 5 = 15$. Pens that are NOT black = $4 + 5 = 9$. Probability = $\frac{9}{15}$, which simplifies to $\frac{3}{5}$. This is a great example of using the complement rule or simple subtraction, similar to logic found in ACT Word Problems Practice Questions with Answers.
2. Answer: $\frac{1}{9}$
   Total outcomes for two dice = $6 \times 6 = 36$. Pairs that sum to 9: (3,6), (4,5), (5,4), (6,3). There are 4 favorable outcomes. Probability = $\frac{4}{36} = \frac{1}{9}$.
3. Answer: $\frac{4}{15}$
   Total students = 30. Using the Principle of Inclusion-Exclusion: Students in band or choir = $12 + 15 - 5 = 22$. Students in neither = $30 - 22 = 8$. Probability = $\frac{8}{30} = \frac{4}{15}$.
4. Answer: $\frac{1}{2}$
   Prime numbers between 1 and 8 are 2, 3, 5, and 7. There are 4 prime numbers out of 8 total sections. Probability = $\frac{4}{8} = \frac{1}{2}$.
5. Answer: $\frac{3}{29}$
   Total coins = 30. First gold coin: $\frac{10}{30} = \frac{1}{3}$. Second gold coin (no replacement): $\frac{9}{29}$. Multiply: $\frac{1}{3} \times \frac{9}{29} = \frac{9}{87} = \frac{3}{29}$.
6. Answer: $\frac{\pi}{150}$
   Garden area = $20 \times 30 = 600$. Fountain radius = 2 (half of diameter). Fountain area = $\pi(2^2) = 4\pi$. Probability = $\frac{4\pi}{600} = \frac{\pi}{150}$. This type of problem often appears in ACT Geometry Practice Questions with Answers.
7. Answer: $\frac{1}{5}$
   Spanish speakers = 20. Those who speak Spanish AND French = 10. Those who speak only Spanish = $20 - 10 = 10$. Total people = 50. Probability = $\frac{10}{50} = \frac{1}{5}$.
8. Answer: $0.65$
   Using the complement rule: $P( \text{not rain}) = 1 - P( \text{rain})$. $1 - 0.35 = 0.65$.
9. Answer: $\frac{3}{100}$
   Total marbles = 10. $P( \text{red}) = \frac{5}{10} = \frac{1}{2}$. $P( \text{green}) = \frac{3}{10}$. $P( \text{blue}) = \frac{2}{10} = \frac{1}{5}$. Multiply: $\frac{1}{2} \times \frac{3}{10} \times \frac{1}{5} = \frac{3}{100}$.
10. Answer: $\frac{28}{91}$
    Total socks = 14. First white sock: $\frac{8}{14} = \frac{4}{7}$. Second white sock: $\frac{7}{13}$. Multiply: $\frac{4}{7} \times \frac{7}{13} = \frac{28}{91}$.

1. A bag contains 4 red candies and 6 green candies. If one is eaten and then a second is chosen, what is the probability both were red?

• A2/15
• B4/25
• C1/10
• D16/100

Frequently Asked Questions

What is the difference between independent and dependent events on the ACT?

Independent events are occurrences where the outcome of the first event does not change the probability of the second, such as rolling a die twice. Dependent events occur when the first outcome changes the total pool for the second, such as drawing cards from a deck without replacing them.

How do I calculate geometric probability for the ACT?

Geometric probability is calculated by taking the area of the "target" region and dividing it by the total area of the entire shape. You should be comfortable calculating areas of circles, rectangles, and triangles, and sometimes using ACT Ratio Practice Questions with Answers strategies to compare them.

What does "without replacement" mean in a probability question?

"Without replacement" indicates that once an item is selected, it is not put back into the group, which reduces both the number of favorable outcomes (if that specific type was picked) and the total number of possible outcomes for the next draw. This creates a dependent probability scenario.

How do Venn diagrams help with probability questions?

Venn diagrams help visualize overlapping sets, allowing you to subtract the "both" category from individual totals to find the unique counts for each group. This prevents double-counting when calculating the probability of "Event A or Event B." For more on managing multiple variables, check out ACT Systems of Equations Practice Questions with Answers.

Can probability on the ACT be greater than 1?

No, a probability can never be greater than 1 or less than 0. If your calculation results in a number like 1.5 or -0.2, you have made an error in your setup, likely by adding probabilities that should have been multiplied or failing to divide by the total outcomes.