Step-by-Step Guidance for Probability and Statistics Practice Problems

Q1. Coupon Redemption Probability

Background

Topic: Binomial Probability Distribution

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

Key Terms and Formulas

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

• Expected Value: $E[X] = np$

• Where:

  • $n$ = number of trials (customers)

  • $k$ = number of successes (purchases)

  • $p$ = probability of success (purchase)

Step-by-Step Guidance

1. Identify the parameters: $n = 12$, $p = 0.3$.

2. For part (a), set $k = 4$ and use the binomial formula to set up $P(X = 4)$.

3. For part (b), "at least 2" means $P(X \geq 2)$. Consider using the complement rule: $P(X \geq 2) = 1 - P(X = 0) - P(X = 1)$.

4. For part (c), use the expected value formula $E[X] = np$ to set up the calculation.

Try solving on your own before revealing the answer!

Q2. Sampling Distribution of the Mean

Background

Topic: Central Limit Theorem and Normal Distribution of Sample Means

This question tests your ability to use the sampling distribution of the sample mean, including calculating probabilities using the normal distribution.

Key Terms and Formulas

• Mean of sample means: $\mu_{\bar{x}} = \mu$

• Standard error: $\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$

• Z-score: $z = \frac{\bar{x} - \mu}{\sigma_{\bar{x}}}$

Step-by-Step Guidance

1. Identify the population mean $\mu = 12$, standard deviation $\sigma = 4$, and sample size $n = 64$.

2. Calculate the standard error: $\sigma_{\bar{x}} = \frac{4}{\sqrt{64}}$.

3. For part (a), find the z-score for $\bar{x} = 13$ and set up the probability $P(\bar{x} > 13)$ using the standard normal table.

4. For part (b), find the z-scores for $\bar{x} = 11$ and $\bar{x} = 13$, then set up $P(11 < \bar{x} < 13)$.

Try solving on your own before revealing the answer!

Q3. Probability with Overlapping Events

Background

Topic: Addition Rule, Conditional Probability, Independence

This question tests your understanding of the addition rule for probabilities, conditional probability, and the concept of independence between events.

Key Terms and Formulas

• Addition Rule: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$

• Conditional Probability: $P(B|A) = \frac{P(A \cap B)}{P(A)}$

• Independence: Events A and B are independent if $P(A \cap B) = P(A)P(B)$

Step-by-Step Guidance

1. For part (a), use the addition rule with $P(\text{online}) = 0.4$, $P(\text{mobile}) = 0.3$, $P(\text{both}) = 0.15$.

2. For part (b), use the conditional probability formula with $P(\text{both})$ and $P(\text{online})$.

3. For part (c), check if $P(\text{both}) = P(\text{online}) \times P(\text{mobile})$ to determine independence.

Try solving on your own before revealing the answer!

Q4. Normal Distribution of Delivery Times

Background

Topic: Normal Distribution, Z-scores, Percentiles

This question tests your ability to use the normal distribution to find probabilities and percentiles for a continuous variable.

Key Terms and Formulas

• Z-score: $z = \frac{x - \mu}{\sigma}$

• Probability: Use the standard normal table to find $P(X < x)$ or $P(a < X < b)$

• Percentile: Find $x$ such that $P(X < x) = p$

Step-by-Step Guidance

1. Identify $\mu = 3.5$ days, $\sigma = 0.8$ days.

2. For part (a), calculate the z-score for $x = 2$ and set up $P(X < 2)$.

3. For part (b), calculate z-scores for $x = 3$ and $x = 5$, then set up $P(3 < X < 5)$.

4. For part (c), find the z-score corresponding to the top 5% (or slowest 5%), then solve for $x$ using $x = \mu + z\sigma$.

Try solving on your own before revealing the answer!

Q5. Normal Distribution of Order Amounts

Background

Topic: Normal Distribution, Z-scores, Percentiles

This question tests your ability to calculate probabilities and percentiles for a normally distributed variable.

Key Terms and Formulas

• Z-score: $z = \frac{x - \mu}{\sigma}$

• Probability: Use the standard normal table for $P(X > x)$ or $P(a < X < b)$

• Percentile: Find $x$ such that $P(X > x) = 0.10$ for the top 10%

Step-by-Step Guidance

1. Identify $\mu = 120$, $\sigma = 25$.

2. For part (a), calculate the z-score for $x = 150$ and set up $P(X > 150)$.

3. For part (b), calculate z-scores for $x = 100$ and $x = 140$, then set up $P(100 < X < 140)$.

4. For part (c), find the z-score for the 90th percentile (top 10%), then solve for $x$ using $x = \mu + z\sigma$.

Try solving on your own before revealing the answer!

Q6. Probability from a Two-Way Table

Background

Topic: Joint, Marginal, and Conditional Probability; Independence

This question tests your ability to read probabilities from a contingency table and determine independence.

Key Terms and Formulas

• Joint Probability: $P(A \cap B)$

• Conditional Probability: $P(A|B) = \frac{P(A \cap B)}{P(B)}$

• Independence: $P(A \cap B) = P(A)P(B)$

Step-by-Step Guidance

1. For part (a), find the joint probability for Returning and Purchased from the table.

2. For part (b), use the conditional probability formula with the appropriate values from the table.

3. For part (c), compare $P(\text{Purchased} | \text{New})$ to $P(\text{Purchased})$ to assess independence.

Try solving on your own before revealing the answer!

Q7. Probability of Sales in Calls (Binomial)

Background

Topic: Binomial Probability Distribution

This question tests your ability to use the binomial distribution to find probabilities and calculate the mean and standard deviation.

Key Terms and Formulas

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

• Mean: $\mu = np$

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

Step-by-Step Guidance

1. Identify $n = 20$, $p = 0.15$.

2. For part (a), set $k = 0$ and use the binomial formula for $P(X = 0)$.

3. For part (b), "more than 3" means $P(X > 3) = 1 - P(X \leq 3)$; set up the sum for $P(X \leq 3)$.

4. For part (c), use the formulas for mean and standard deviation.

Try solving on your own before revealing the answer!

Q8. Probability with Conditional Events

Background

Topic: Multiplication Rule, Conditional Probability, Complements

This question tests your understanding of joint probability, conditional probability, and complements.

Key Terms and Formulas

• Joint Probability: $P(A \cap B) = P(A) \times P(B|A)$

• Conditional Probability: $P(B|A) = \frac{P(A \cap B)}{P(A)}$

• Complement: $P(\text{not } A) = 1 - P(A)$

Step-by-Step Guidance

1. For part (a), multiply the probability an order is returned by the probability it is damaged given it was returned.

2. For part (b), use the conditional probability formula with the given values.

3. For part (c), use the complement rule to find the probability an order is not returned.

Try solving on your own before revealing the answer!