12-1 Additional Practice Probability Events Answer Key

12-1 Additional Practice: Probability Events – Answers & Explanations

This comprehensive guide provides detailed answers and explanations for a hypothetical "12-1 Additional Practice" worksheet focusing on probability events. While a specific worksheet isn't provided, this article covers a wide range of probability concepts and problem types you're likely to encounter at this level. We'll tackle everything from basic probability calculations to more complex scenarios involving dependent and independent events, conditional probability, and combinations.

Remember, understanding the underlying principles is crucial. Don't just memorize formulas; focus on grasping the logic behind them.

Understanding Basic Probability

Before diving into the problems, let's solidify our understanding of fundamental probability concepts.

Probability Definition

Probability measures the likelihood of an event occurring. It's expressed as a number between 0 and 1, inclusive.

• 0: Represents an impossible event.
• 1: Represents a certain event.
• 0.5 (or 1/2): Represents an event with an equal chance of occurring or not occurring.

The formula for probability is:

P(A) = (Number of favorable outcomes) / (Total number of possible outcomes)

Where P(A) denotes the probability of event A.

Types of Events

• Independent Events: The outcome of one event doesn't affect the outcome of another. For example, flipping a coin twice.
• Dependent Events: The outcome of one event does affect the outcome of another. For example, drawing two cards from a deck without replacement.
• Mutually Exclusive Events: Two events that cannot occur at the same time. For example, rolling a die and getting a 1 and getting a 6 simultaneously.

Tackling Probability Problems: A Step-by-Step Approach

Let's work through several example problems, mirroring the types you might find in a "12-1 Additional Practice" section.

Problem 1: Simple Probability

A bag contains 5 red marbles, 3 blue marbles, and 2 green marbles. What is the probability of randomly selecting a red marble?

Solution:

1. Identify favorable outcomes: There are 5 red marbles.
2. Identify total possible outcomes: There are 5 + 3 + 2 = 10 marbles in total.
3. Apply the probability formula: P(Red) = 5/10 = 1/2 = 0.5

Therefore, the probability of selecting a red marble is 0.5 or 50%.

Problem 2: Probability with Independent Events

You flip a fair coin twice. What is the probability of getting heads on both flips?

Solution:

1. Independent events: Each coin flip is independent of the other.
2. Probability of heads on one flip: P(Heads) = 1/2
3. Probability of heads on two flips: Since the events are independent, we multiply their probabilities: P(Heads, Heads) = (1/2) * (1/2) = 1/4 = 0.25

Therefore, the probability of getting heads on both flips is 0.25 or 25%.

Problem 3: Probability with Dependent Events

A box contains 4 red balls and 6 blue balls. You draw two balls without replacement. What is the probability that both balls are red?

Solution:

1. Dependent events: The outcome of the first draw affects the outcome of the second draw.
2. Probability of the first ball being red: P(Red1) = 4/10 = 2/5
3. Probability of the second ball being red, given the first was red: After drawing one red ball, there are 3 red balls and 6 blue balls left (9 balls total). P(Red2 | Red1) = 3/9 = 1/3
4. Probability of both balls being red: Multiply the probabilities: P(Red1 and Red2) = (2/5) * (1/3) = 2/15

Therefore, the probability that both balls are red is 2/15.

Problem 4: Conditional Probability

A company produces light bulbs. 90% are non-defective, while 10% are defective. Of the non-defective bulbs, 95% are bright, and 5% are dim. What is the probability that a randomly selected bulb is both non-defective and bright?

Solution:

This involves conditional probability. We want to find P(Bright | Non-defective).

1. Probability of a non-defective bulb: P(Non-defective) = 0.9
2. Probability of a bright bulb given it's non-defective: P(Bright | Non-defective) = 0.95
3. Probability of a non-defective and bright bulb: P(Non-defective and Bright) = P(Non-defective) * P(Bright | Non-defective) = 0.9 * 0.95 = 0.855

Therefore, the probability that a randomly selected bulb is both non-defective and bright is 0.855 or 85.5%.

Problem 5: Probability using Combinations (Permutation)

A committee of 3 people is to be selected from a group of 5 people. How many different committees are possible?

Solution:

This problem uses combinations because the order in which the committee members are selected doesn't matter. The formula for combinations is:

nCr = n! / (r! * (n-r)!)

Where:

• n is the total number of items (5 people)
• r is the number of items to choose (3 people)
• ! denotes the factorial (e.g., 5! = 5 * 4 * 3 * 2 * 1)

5C3 = 5! / (3! * 2!) = (5 * 4 * 3 * 2 * 1) / ((3 * 2 * 1) * (2 * 1)) = 10

Therefore, there are 10 different possible committees.

Problem 6: Mutually Exclusive Events

A single six-sided die is rolled. What is the probability of rolling a number less than 3 OR rolling an even number?

Solution:

1. Numbers less than 3: {1, 2} Probability = 2/6 = 1/3
2. Even numbers: {2, 4, 6} Probability = 3/6 = 1/2
3. Overlap: The number 2 is in both sets. To avoid double-counting, we use the inclusion-exclusion principle:

P(A or B) = P(A) + P(B) - P(A and B)

P(Less than 3 OR Even) = (1/3) + (1/2) - (1/6) = 4/6 = 2/3

Therefore, the probability of rolling a number less than 3 OR an even number is 2/3.

Advanced Probability Concepts (Beyond Basic 12-1)

While a typical "12-1 Additional Practice" might not cover these, let's briefly touch upon more advanced topics for broader understanding.

• Bayes' Theorem: Used to update probabilities based on new evidence. It's particularly useful in situations with conditional probabilities.
• Expected Value: The average outcome you'd expect over many repetitions of an experiment.
• Probability Distributions: Describing the probability of different outcomes (e.g., binomial distribution, normal distribution).
• Law of Large Numbers: As the number of trials increases, the observed frequency of an event will approach its theoretical probability.

Tips for Mastering Probability

• Practice Regularly: The more problems you solve, the better your understanding will become.
• Visualize Problems: Draw diagrams, charts, or use other visual aids to help you understand the problem scenarios.
• Break Down Complex Problems: Divide complex problems into smaller, more manageable parts.
• Check Your Work: Carefully review your calculations to minimize errors.
• Seek Help When Needed: Don't hesitate to ask for clarification from your teacher or tutor if you're struggling.

This comprehensive guide provides a solid foundation for tackling probability problems. Remember that consistent practice and a deep understanding of the underlying principles are key to mastering this important area of mathematics. While this addresses a hypothetical "12-1 Additional Practice," the concepts and problem-solving strategies are broadly applicable to many probability exercises. Use this as a resource to bolster your understanding and tackle any probability challenge with confidence!