Precalculus Probability and Counting Principles Study Guide

Q1. Probability of Students Wearing Socks

Background

Topic: Binomial Probability

This question tests your understanding of the binomial probability distribution, which models the probability of a fixed number of successes in a fixed number of independent trials, each with the same probability of success.

Key Terms and Formulas

• Binomial Probability Formula: $P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$

• n = number of trials (students chosen)

• k = number of successes (students wearing socks)

• p = probability of success (student wears socks)

Step-by-Step Guidance

1. Identify the values: $n = 10$, $p = 0.85$ (probability a student wears socks).

2. For each part (a–d), determine the value of $k$ (number of students wearing socks) or the range of $k$ values.

3. Set up the binomial probability formula for each scenario:

   • (a) All wearing socks: $k = 10$

   • (b) None wearing socks: $k = 0$

   • (c) At least 7 wearing socks: $k = 7, 8, 9, 10$ (sum probabilities for these values)

   • (d) Exactly 1 wearing socks: $k = 1$

4. For (c), remember to add the probabilities for each value from 7 to 10.

Try solving on your own before revealing the answer!

Q2. Minimum Number of Sensors for Desired Certainty

Background

Topic: Binomial Probability and Cumulative Probability

This question asks you to determine the minimum number of sensors needed so that the probability of at least one detecting movement exceeds 97%.

Key Terms and Formulas

• Probability at least one detects: $P(\text{at least one}) = 1 - P(\text{none detect})$

• Probability none detect: $(1-p)^n$

• p = probability a sensor detects movement (0.80)

• n = number of sensors (what you are solving for)

Step-by-Step Guidance

1. Express the probability that at least one sensor detects movement: $P = 1 - (1-p)^n$.

2. Set up the inequality: $1 - (1-0.8)^n > 0.97$.

3. Rearrange to solve for $n$.

4. Take the logarithm of both sides to isolate $n$.

Try solving on your own before revealing the answer!

Q3. Tim's Free-Throw Probability and Distribution

Background

Topic: Binomial Distribution – Mean, Standard Deviation, and Probability Histogram

This question involves calculating the expected value (mean), standard deviation, and probability distribution for a binomial experiment.

Key Terms and Formulas

• Mean: $\mu = np$

• Standard deviation: $\sigma = \sqrt{np(1-p)}$

• Probability for $k$ successes: $P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$

• n = 8, p = 0.82

Step-by-Step Guidance

1. Calculate the mean: $\mu = n \times p$.

2. Calculate the standard deviation: $\sigma = \sqrt{n p (1-p)}$.

3. For the histogram, compute $P(X = k)$ for $k = 0, 1, ..., 8$ using the binomial formula.

4. Sketch or list the probabilities for each value of $k$ to form the histogram.

Try solving on your own before revealing the answer!

Q4. Comparing Probabilities for Tim's Free Throws

Background

Topic: Cumulative Binomial Probability

This question asks you to compare the probability of Tim making 6 or fewer shots versus 7 or more out of 8 attempts.

Key Terms and Formulas

• Cumulative probability: $P(X \leq k)$ and $P(X \geq k)$

• Use the binomial probability formula for each value and sum as needed.

Step-by-Step Guidance

1. Calculate $P(X \leq 6)$ by summing probabilities for $k = 0$ to $6$.

2. Calculate $P(X \geq 7)$ by summing probabilities for $k = 7$ and $8$.

3. Compare the two probabilities to determine which is more likely.

Try solving on your own before revealing the answer!

Q5. Criteria of a Binomial Experiment

Background

Topic: Binomial Experiment Properties

This question tests your understanding of the defining characteristics of a binomial experiment.

Key Terms

• Binomial experiment: An experiment that meets specific criteria for binomial probability.

Step-by-Step Guidance

1. Recall the properties: fixed number of trials, only two possible outcomes per trial, etc.

2. List each criterion clearly and concisely.

3. Think about why each criterion is necessary for the binomial model to apply.

Try solving on your own before revealing the answer!

Q6. Probability of Winning a Board Game with Dice

Background

Topic: Probability with Dice and Binomial Probability

This question involves finding the probability of a specific event with dice, and then using the binomial model to find the probability of that event occurring multiple times in repeated trials.

Key Terms and Formulas

• Probability of a single event with dice (counting outcomes)

• Binomial probability for repeated independent events

Step-by-Step Guidance

1. List all possible outcomes for rolling two dice and adding the numbers.

2. Count the number of outcomes that result in a sum of 8 or at least one die showing a 6.

3. Calculate the probability of winning in one roll.

4. Set up the binomial probability formula for winning 3 times out of 8 rolls using the probability from the previous step.

Try solving on your own before revealing the answer!

Q7. Probability of Accepting a Yogurt Shipment

Background

Topic: Binomial Probability – Quality Control

This question asks you to find the probability that, in a sample of 16 yogurts, no more than 2 have open lids, given a 95% success rate for closed lids.

Key Terms and Formulas

• n = 16, p = 0.95 (probability lid is on)

• Probability of at most 2 open lids: sum probabilities for 0, 1, and 2 open lids

• Use the binomial probability formula for each value of k (number of open lids)

Step-by-Step Guidance

1. Let k = number of open lids (so probability of open lid is 0.05).

2. Calculate $P(X = 0)$, $P(X = 1)$, and $P(X = 2)$ using the binomial formula.

3. Add these probabilities to find the total probability of accepting the shipment.

Try solving on your own before revealing the answer!

Q8. Probability of Passing a Multiple Choice Test by Guessing

Background

Topic: Binomial Probability – Multiple Choice Tests

This question asks you to find the probability of getting at least 6 correct answers out of 10 by guessing, with 4 choices per question.

Key Terms and Formulas

• n = 10, p = 0.25 (probability of guessing correctly)

• Probability of at least 6 correct: sum probabilities for k = 6, 7, 8, 9, 10

• Use the binomial probability formula for each value of k

Step-by-Step Guidance

1. Set up the binomial probability formula for each value of k from 6 to 10.

2. Calculate each probability and add them together to find the total probability of passing.

Try solving on your own before revealing the answer!

Chapter 4 Q1. Counting Outcomes (Combinations and Permutations)

Background

Topic: Counting Principles – Combinations and Permutations

This question tests your ability to distinguish between combinations (order does not matter) and permutations (order matters) and to use the appropriate formulas.

Key Terms and Formulas

• Combinations: $C(n, r) = \frac{n!}{r!(n-r)!}$

• Permutations: $P(n, r) = \frac{n!}{(n-r)!}$

Step-by-Step Guidance

1. For each part, determine if order matters (permutation) or not (combination).

2. Set up the appropriate formula for each scenario:

   • (a) Choosing jewelry: combination

   • (b) Selecting students: combination

   • (c) Assigning prizes: permutation

   • (d) Arranging posters: permutation

3. Plug in the values for n and r for each part and simplify the expressions.

Try solving on your own before revealing the answer!

Chapter 4 Q2. Probability with Letters in "CAULIFLOWER"

Background

Topic: Basic Probability with Counting

This question involves finding probabilities by counting favorable outcomes and dividing by the total number of possible outcomes.

Key Terms and Formulas

• Probability: $P(E) = \frac{\text{number of favorable outcomes}}{\text{total number of outcomes}}$

• Combinations for drawing two cards: $C(n, 2)$

Step-by-Step Guidance

1. Count the total number of letters in "CAULIFLOWER".

2. For each part, count the number of favorable outcomes (e.g., vowels, L's, etc.).

3. For two cards, use combinations to count the number of ways to draw two vowels, two consonants, or one of each.

Try solving on your own before revealing the answer!

Chapter 4 Q3. Venn Diagram Probability

Background

Topic: Venn Diagrams and Probability

This question asks you to use a Venn diagram to find probabilities involving unions, intersections, and complements of sets.

Key Terms and Formulas

• Union: $P(A \cup B)$

• Intersection: $P(A \cap B)$

• Complement: $P(\sim A)$

• Conditional probability: $P(A|B)$

Step-by-Step Guidance

1. For each part, identify which regions of the Venn diagram correspond to the event described.

2. Add the numbers in the relevant regions to find the numerator (favorable outcomes).

3. Find the total number of students by adding all numbers in the diagram.

4. Set up the probability as a fraction: $P(E) = \frac{\text{favorable outcomes}}{\text{total outcomes}}$.

5. For conditional probability, use $P(A|B) = \frac{P(A \cap B)}{P(B)}$.

Try solving on your own before revealing the answer!