Step-by-Step Guidance for Assignment 2: Probability and Statistics Problems

Q1. University Survey: Probability Calculations from a Contingency Table

Background

Topic: Probability using contingency tables

This question tests your ability to calculate probabilities (simple, joint, conditional, and compound) from a two-way table, and to reason about mutually exclusive and independent events.

Key Terms and Formulas

• Probability: $P(A) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}$

• Joint Probability: $P(A \cap B) = \frac{\text{Number in both A and B}}{\text{Total}}$

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

• Mutually Exclusive Events: Events that cannot occur at the same time.

• Independent Events: $P(A \cap B) = P(A) \times P(B)$

Step-by-Step Guidance

1. Carefully read the table and identify the totals for each category (faculty, students, support, oppose, neutral, and overall total).

2. For each probability, determine the numerator (number of individuals fitting the criteria) and the denominator (total surveyed, which is 5000).

3. For joint probabilities (e.g., student and opposes), find the cell in the table where both conditions are met.

4. For 'or' probabilities (e.g., faculty or supports), remember to use the addition rule: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$.

5. For conditional probabilities, use the formula $P(A|B) = \frac{P(A \cap B)}{P(B)}$ and identify the correct numerator and denominator from the table.

6. For mutually exclusive and independence questions, use the definitions and check if the intersection is empty (mutually exclusive) or if $P(A \cap B) = P(A) \times P(B)$ (independent).

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Q2. Shirts Sold: Probability from a Frequency Table

Background

Topic: Empirical (relative frequency) probability

This question asks you to calculate probabilities based on observed frequencies and to identify the type of probability used.

Key Terms and Formulas

• Relative Frequency Probability: $P(A) = \frac{\text{Frequency of A}}{\text{Total frequency}}$

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

Step-by-Step Guidance

1. Sum the frequencies for all shirt sizes to confirm the total number of shirts sold.

2. For the probability of a large shirt, use the frequency for 'Large' divided by the total.

3. For the probability of not buying a small shirt, sum the frequencies for all sizes except 'Small' and divide by the total, or use the complement rule.

4. For the concept type, recall the three types of probability: classical, empirical (relative frequency), and subjective. Decide which applies here and explain why.

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Q3. Jobs-for-All: Binomial Probability Calculations

Background

Topic: Binomial probability

This question involves calculating the probability of a certain number of successes in a fixed number of independent trials, each with the same probability of success.

Key Terms and Formulas

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

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

• Where $n$ = number of trials, $k$ = number of successes, $p$ = probability of success.

Step-by-Step Guidance

1. Identify $n = 3$ (number of people contacted), $p = 0.9$ (probability of finding a job), and $q = 1 - p = 0.1$ (probability of not finding a job).

2. For all three finding jobs, use $k = 3$ in the binomial formula.

3. For none finding jobs, use $k = 0$ in the binomial formula.

4. For at least one finding a job, use the complement rule: $1 - P(\text{none})$.

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Q4. Curling Competition: Counting Arrangements (Permutations)

Background

Topic: Permutations

This question asks you to count the number of ways to assign gold, silver, and bronze medals to 20 teams, where order matters.

Key Terms and Formulas

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

• Where $n$ = total items, $r$ = number of positions to fill.

Step-by-Step Guidance

1. Identify $n = 20$ (teams) and $r = 3$ (medals: gold, silver, bronze).

2. Plug these values into the permutation formula to set up the calculation.

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Q5. Lottery: Probability of Winning with Combinations

Background

Topic: Combinatorics and probability

This question involves calculating the probability of selecting the correct 3 numbers (in any order) out of 25 possible numbers.

Key Terms and Formulas

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

• Probability: $P = \frac{1}{\text{Total number of possible combinations}}$

Step-by-Step Guidance

1. Identify $n = 25$ (total numbers) and $r = 3$ (numbers to choose).

2. Calculate the total number of ways to choose 3 numbers from 25 using the combination formula.

3. Set up the probability as $P = \frac{1}{\text{number of combinations}}$.

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Q6. Major Motors: Probability Tree and Compound Probability

Background

Topic: Probability trees, compound probability, and independence

This question asks you to draw a probability tree, calculate compound probabilities, and test for independence between two events.

Key Terms and Formulas

• Probability Tree: A diagram showing all possible outcomes and their probabilities.

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

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

Step-by-Step Guidance

1. Draw a tree diagram with two branches: new car (60%) and used car (40%).

2. For each branch, add sub-branches for purchasing an extended warranty (50% for new, 75% for used) and not purchasing.

3. To find the probability of new car and extended warranty, multiply the probabilities along that path.

4. To find the probability of purchasing an extended warranty (regardless of car type), sum the probabilities of both relevant branches.

5. To test independence, compare $P(\text{new car} \cap \text{warranty})$ with $P(\text{new car}) \times P(\text{warranty})$.

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