Q1. Find for the discrete random variable given the probability distribution:

Background

Topic: Probability Distribution for Discrete Random Variables

This question tests your understanding of probability distributions and how to find missing probabilities when the total must sum to 1.

Key Terms and Formulas:

• Discrete random variable: Takes on specific, countable values.

• Probability distribution: A table showing the probabilities for each possible value of .

• Sum of probabilities:

Step-by-Step Guidance

1. Write out all the given probabilities for : , , , , , .

2. Recall that the sum of all probabilities must equal 1: .

3. Add up all the known probabilities.

4. Set up the equation to solve for by subtracting the sum of the known probabilities from 1.

Try solving on your own before revealing the answer!

Q2. Probability of customers buying another Micron PC (binomial probability)

Background

Topic: Binomial Probability

This question tests your ability to use the binomial probability formula to calculate probabilities for a fixed number of trials with two possible outcomes (success/failure).

Key Terms and Formulas:

• Binomial probability formula:

• = number of trials (here, 4 customers)

• = probability of success (here, 0.88)

• = number of successes

Step-by-Step Guidance

1. For part (a), identify , , (all four buy again).

2. Plug these values into the binomial formula: .

3. For part (b), (none buy again): .

4. For part (c), (exactly one buys again): .

5. Set up each calculation, but stop before computing the final values.

Try solving on your own before revealing the answer!

Q3. Find , , and ; Are A and B independent?

Background

Topic: Probability Rules and Independence

This question tests your understanding of probability rules for unions, intersections, complements, and independence.

Key Terms and Formulas:

• Events are independent if

Step-by-Step Guidance

1. Write down the given probabilities: , , .

2. Calculate using the formula: .

3. Calculate using .

4. For independence, compare to .

5. Set up the calculations, but stop before the final numeric answers.

Try solving on your own before revealing the answer!

Q4. Probability calculations for events A and B in a sample space of dice rolls

Background

Topic: Probability, Sample Space, Conditional Probability, Mutual Exclusivity, Independence

This question tests your ability to calculate probabilities for events, unions, intersections, conditional probabilities, and to determine mutual exclusivity and independence.

Key Terms and Formulas:

• Sample space: All possible outcomes (here, 1–6).

• : Probability of event A.

• : Probability of event B.

• : Probability of A or B.

• : Probability of both A and B.

• Conditional probability:

• Mutually exclusive:

• Independent:

Step-by-Step Guidance

1. List the elements of A and B: , .

2. Calculate : Number of elements in A divided by total outcomes.

3. Calculate : Number of elements in B divided by total outcomes.

4. Find and .

5. Set up and using the conditional probability formula.

6. Check if is empty for mutual exclusivity, and compare to for independence.

Try solving on your own before revealing the answer!

Q5. Probability of drawing two cards: both aces, or same value

Background

Topic: Probability with Combinatorics

This question tests your ability to calculate probabilities using combinations for drawing cards from a deck.

Key Terms and Formulas:

• Number of ways to choose 2 cards:

• Number of ways to choose 2 aces:

• Number of ways to choose 2 cards of same value: , then multiply by 13 values.

• Probability:

Step-by-Step Guidance

1. Calculate total ways to choose 2 cards: .

2. For both aces: ways to choose 2 aces.

3. Set up probability for both aces: .

4. For same value: For each value, ways; there are 13 values.

5. Set up probability for same value: .

Try solving on your own before revealing the answer!

Q6. Probability for lifetime of electrical part (normal distribution)

Background

Topic: Normal Distribution, Probability Calculations

This question tests your ability to use the normal distribution to find probabilities and percentiles.

Key Terms and Formulas:

• Mean (): 100 hours

• Standard deviation (): 20 hours

• Z-score:

• Probability: Use standard normal table for

• Percentile: Find such that

Step-by-Step Guidance

1. For each part, calculate the -score for the relevant value(s).

2. For part (a):

3. For part (b): Find for 98 and 104, then calculate .

4. For part (c): , then find .

5. For part (d): Find corresponding to the 99th percentile, then solve for .

Try solving on your own before revealing the answer!

Q7. Probability for number of Hondas made in North America (binomial distribution)

Background

Topic: Binomial Probability

This question tests your ability to use the binomial distribution for probabilities involving a fixed number of trials and a given probability of success.

Key Terms and Formulas:

• Binomial probability:

• ,

• For cumulative probabilities, sum over relevant values.

Step-by-Step Guidance

1. For part (a): Calculate by summing for to $13$.

2. For part (b): Calculate by summing for to $13$.

3. For part (c): Calculate using the binomial formula.

4. Set up the calculations, but stop before computing the final values.

Try solving on your own before revealing the answer!

Q8. Construct a 90% confidence interval for the proportion of likely voters for GOP candidate

Background

Topic: Confidence Interval for Proportion

This question tests your ability to construct a confidence interval for a population proportion using sample data.

Key Terms and Formulas:

• Sample proportion:

• Standard error:

• Critical value for 90% confidence:

• Confidence interval:

Step-by-Step Guidance

1. Calculate sample proportion: .

2. Calculate standard error: .

3. Find the critical value for 90% confidence ().

4. Set up the confidence interval formula: .

5. Interpret the interval in context, but stop before calculating the final interval.

Try solving on your own before revealing the answer!

Q9. Human body temperature: percentage considered fever, and cutoff for top 5%

Background

Topic: Normal Distribution, Percentiles

This question tests your ability to use the normal distribution to find probabilities and cutoff values.

Key Terms and Formulas:

• Mean (): 98.2°F

• Standard deviation (): 0.62°F

• Z-score:

• Use standard normal table for probabilities

Step-by-Step Guidance

1. For part (a): Calculate for 100.6°F.

2. Find using the standard normal table.

3. For part (b): Find such that (top 5%).

4. Use the inverse normal table to find the cutoff temperature.

Try solving on your own before revealing the answer!

Q10. Probability for sample proportion of Cat People (normal approximation)

Background

Topic: Sampling Distribution of Proportion, Normal Approximation

This question tests your ability to use the normal approximation for the sampling distribution of a proportion.

Key Terms and Formulas:

• Population proportion:

• Sample size:

• Standard error:

• Z-score:

• Use normal table for probabilities

Step-by-Step Guidance

1. Calculate .

2. For part (a): Find for .

3. For part (b): Find for .

4. Use the normal table to find the probabilities for each .

Try solving on your own before revealing the answer!

Q11. Definitions: Sample space, mutually exclusive events, independent events, complement, confidence level, critical value, margin of error, standard error, expected value and variance, continuous/discrete random variable/distribution

Background

Topic: Fundamental Concepts in Probability and Statistics

This question tests your understanding of key definitions and concepts in statistics.

Key Terms and Formulas:

• Sample space: Set of all possible outcomes.

• Mutually exclusive events: Events that cannot occur together.

• Independent events: Events where occurrence of one does not affect the other.

• Complement: is the event that A does not occur.

• Confidence level: Probability that the interval contains the true parameter.

• Critical value: Value from the distribution corresponding to the desired confidence.

• Margin of error:

• Standard error: Standard deviation of a sampling distribution.

• Expected value:

• Variance:

• Continuous random variable: Can take any value in an interval.

• Discrete random variable: Takes countable values.

• Continuous probability distribution: Probability density function.

• Discrete probability distribution: Probability mass function.

Step-by-Step Guidance

1. Review each definition and formula listed above.

2. Write out your own examples for each term to reinforce understanding.

3. For expected value and variance, practice with a simple probability distribution.

4. For continuous/discrete variables, think of real-world examples.

Try writing your own definitions and examples before revealing the answer!