PSTAT 5A Practice Worksheet 3

Comprehensive Review: Probability, Counting, an Conditional Probability

Author

Student Name: ________________________

Published

July 9, 2025

Instructions and Overview

⏰ Time Allocation:

  • Section A (Warm-up): 8 minutes

  • Section B (Intermediate): 15 minutes

  • Section C (Advanced): 12 minutes

  • Section D (Review): 15 minutes

  • Total: 50 minutes

📝 Important Instructions:

  • Use the formulas provided for guidance

  • Round final answers to 4 decimal places unless otherwise specified

  • Identify your approach before calculating

  • Use calculator as needed

📚 Key Formulas Reference:

Basic Probability:

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

  • Law of Total Probability: \(P(A) = \sum P(A|B_i) \cdot P(B_i)\)

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

  • Multiplication Rule: \(P(A \cap B) = P(A) \cdot P(B|A) = P(B) \cdot P(A|B)\)

Counting:

  • Multiplication Rule: If a procedure consists of \(k\) steps, with \(n_1\) ways for step 1, \(n_2\) for step 2, …, \(n_k\) for step \(k\), then total ways: \(n_1 \times n_2 \times \cdots \times n_k\)

  • Factorial: \(n! = n \times (n-1) \times (n-2) \times \cdots \times 2 \times 1\)

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

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

Section A: Probability

⏱️ Estimated time: 8 minutes

Problem A1: Probability Distributions

Each row in the table below is a proposed grade distribution for a class. Identify each as a valid or invalid probability distribution, and explain your reasoning.

Class A B C D F
(a) 0.3 0.3 0.3 0.2 0.1
(b) 0 0 1 0 0
(c) 0.3 0.3 0.3 0 0
(d) 0.3 0.5 0.2 0.1 -0.1
(e) 0.2 0.4 0.2 0.1 0.1
(f) 0 -0.1 1.1 0 0

Work Space:







Section B: Permutations and Combination

⏱️ Estimated time: 15 minutes

Problem B1: Permutations and Combinations

A cybersecurity team needs to create a secure access protocol.

Part (a): How many 6-character passwords can be formed using 3 specific letters and 3 specific digits if repetitions are not allowed and letters must come before digits?

Tip

Since letters must come before digits, think of this as two separate arrangement problems:

  • First, arrange the 3 letters in the first 3 positions

  • Then, arrange the 3 digits in the last 3 positions

  • Use the multiplication principle to combine these results

Part (b): If the team wants to select 4 people from 12 employees to form a security committee where order doesn’t matter, how many ways can this be done?

Tip

Since order doesn’t matter, this is a combination problem. Ask yourself:

  • Are we arranging people in specific positions, or just selecting a group?

  • Which formula should you use: \(P(n,r)\) or \(C(n,r)\)?

Work Space:







Section C: Conditional Probability

⏱️ Estimated time: 12 minutes

Problem C1: Drawing Cards (Without Replacement)

You draw two cards, one after the other, from a standard 52-card deck without putting the first card back. Let

A = \(\{\text{“first card is a heart”}\}\),

B = \(\{\text{“second card is an ace”}\}\).

  1. P(A)

  2. \(P\bigl(A\text{ and }B\bigr)\)

  3. \(P\bigl(B\mid A\bigr)\)

  4. P(B)

  5. Compare your answers in (3) vs. (4). Why are they different (or the same)? What does this tell you about drawing cards without replacement?

Work Space:











Section D: Conditional Probability

⏱️ Estimated time: 15 minutes

Problem D1: Advanced Counting with Restrictions

A restaurant offers a prix fixe menu where customers must choose:

  • 1 appetizer from 6 options

  • 1 main course from 8 options

  • 1 dessert from 5 options

However, there are restrictions:

  • If you choose the seafood appetizer, you cannot choose the vegetarian main course

  • If you choose the chocolate dessert, you must choose either the beef or chicken main course (3 of the 8 main courses)

Part (a): How many valid meal combinations are possible?

Part (b): If customers choose randomly among valid combinations, what is the probability someone chooses the chocolate dessert?

Work Space: