Guided Practice
Work through probability problems step-by-step with detailed explanations and hints.
Learn
In this lesson, you will practice applying both theoretical and experimental probability concepts through guided examples. Each problem includes step-by-step hints to help you develop problem-solving strategies.
Key Problem-Solving Steps
- Identify the sample space: List all possible outcomes
- Identify favorable outcomes: Determine which outcomes match the event
- Calculate the probability: Use P(event) = favorable outcomes / total outcomes
- Simplify and interpret: Express as a fraction, decimal, or percent
Remember These Probability Rules
- Probability is always between 0 and 1 (or 0% to 100%)
- P(impossible event) = 0
- P(certain event) = 1
- P(not A) = 1 - P(A)
Examples
Example 1: Spinner Probability
Problem: A spinner is divided into 8 equal sections numbered 1 through 8. What is the probability of landing on an even number?
Step 1: Find the sample space
The sample space is {1, 2, 3, 4, 5, 6, 7, 8}. Total outcomes = 8
Step 2: Identify favorable outcomes
Even numbers in the sample space: {2, 4, 6, 8}. Favorable outcomes = 4
Step 3: Calculate probability
P(even) = 4/8 = 1/2 = 0.5 = 50%
Example 2: Card Drawing
Problem: A bag contains 3 red marbles, 5 blue marbles, and 2 green marbles. What is the probability of drawing a marble that is NOT blue?
Step 1: Find the total
Total marbles = 3 + 5 + 2 = 10
Step 2: Use the complement rule
P(not blue) = 1 - P(blue) = 1 - 5/10 = 1 - 1/2 = 1/2
OR count non-blue: 3 red + 2 green = 5, so P(not blue) = 5/10 = 1/2
Practice
Try these guided practice problems. Use the hints if you get stuck.
Problem 1
A standard die is rolled. What is the probability of rolling a number greater than 4?
Hint
Numbers greater than 4 are 5 and 6. There are 6 possible outcomes total.
Answer
P(greater than 4) = 2/6 = 1/3
Problem 2
A coin is flipped 3 times. What is the probability of getting exactly 2 heads?
Hint
List all possible outcomes: HHH, HHT, HTH, THH, HTT, THT, TTH, TTT (8 total). Count those with exactly 2 heads.
Answer
Outcomes with 2 heads: HHT, HTH, THH (3 outcomes). P(exactly 2 heads) = 3/8
Problem 3
A jar contains 4 red, 6 yellow, and 5 orange jellybeans. What is the probability of picking a red or orange jellybean?
Hint
Add the favorable outcomes (red + orange) and divide by total.
Answer
P(red or orange) = (4 + 5) / 15 = 9/15 = 3/5
Problem 4
In a class of 30 students, 12 play soccer, 8 play basketball, and 10 play neither. What is the probability that a randomly selected student plays at least one sport?
Hint
Use the complement: P(at least one) = 1 - P(neither)
Answer
P(at least one sport) = 1 - 10/30 = 1 - 1/3 = 2/3
Problem 5
A spinner has 5 equal sections: 2 red, 2 blue, and 1 green. If you spin twice, what is the probability of landing on green at least once?
Hint
P(green at least once) = 1 - P(no green in 2 spins). P(not green) = 4/5 for each spin.
Answer
P(no green) = (4/5) x (4/5) = 16/25. P(at least one green) = 1 - 16/25 = 9/25
Problem 6
A deck has 10 cards numbered 1-10. Two cards are drawn without replacement. What is the probability that both cards are even?
Hint
Even cards: 2, 4, 6, 8, 10 (5 cards). First draw: 5/10. After removing one even card: 4/9.
Answer
P(both even) = (5/10) x (4/9) = 20/90 = 2/9
Problem 7
A weather forecast shows 40% chance of rain on Saturday and 60% chance on Sunday. Assuming independence, what is the probability of rain on both days?
Hint
For independent events, multiply probabilities: P(A and B) = P(A) x P(B)
Answer
P(rain both days) = 0.40 x 0.60 = 0.24 = 24%
Problem 8
In an experiment, a coin was flipped 50 times and landed on heads 28 times. What is the experimental probability of heads? How does this compare to the theoretical probability?
Hint
Experimental probability = observed outcomes / total trials. Theoretical probability of heads = 1/2 = 0.5
Answer
Experimental P(heads) = 28/50 = 0.56 = 56%. This is 6% higher than the theoretical probability of 50%. With more trials, we would expect the experimental probability to get closer to 50%.
Check Your Understanding
Answer these questions to verify your mastery of probability problem-solving.
- What are the four main steps for solving basic probability problems?
- When would you use the complement rule (1 - P(A)) instead of counting favorable outcomes directly?
- What is the difference between theoretical and experimental probability?
- If P(A) = 0.3 and P(B) = 0.5, and A and B are independent, what is P(A and B)?
Next Steps
- Review any problems where you needed hints
- Try creating your own probability problems using similar scenarios
- Continue to Word Problems for real-world applications