PSTAT 5A Practice Worksheet 3

Comprehensive Review: Probability, Counting, an Conditional Probability

Author

Student Name: ________________________

Published

July 29, 2025

Instructions and Overview

⏰ Time Allocation:

Section A (Warm-up): 8 minutes
Section B (Intermediate): 15 minutes
Section C (Advanced): 15 minutes
Section D (Review): 12 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: $P(A|B) = \frac{P(A \cap B)}{P(B)}$
Bayes’ Theorem: $P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)}$
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:

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: 15 minutes

Problem B1: Conditional Probability and Medical Testing

A new COVID variant test has the following characteristics:

The variant affects 3% of the tested population
The test correctly identifies 95% of people with the variant (sensitivity)
The test correctly identifies 92% of people without the variant (specificity)

Part (a): What is the probability that a randomly selected person tests positive?

Part (b): If someone tests positive, what is the probability they actually have the variant?

Part (c): If someone tests negative, what is the probability they actually don’t have the variant?

Part (d) [Challenge]: The health department wants to reduce false positives. They decide to require two consecutive positive tests for a positive diagnosis. Assuming test results are independent, what is the new probability that someone with two positive tests actually has the variant?

Work Space:

Section C: Conditional Probability

⏱️ Estimated time: 15 minutes

Problem C1: 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:

Section D: Review

⏱️ Estimated time: 12 minutes

Problem B3: Daily Expenses

Sally gets a cup of coffee and a muffin every day for breakfast from one of the many coffee shops in her neighborhood. She picks a coffee shop each morning at random and independently of previous days. The average price of a cup of coffee is $1.40 with a standard deviation of 30¢ ($0.30), the average price of a muffin is $2.50 with a standard deviation of 15¢, and the two prices are independent of each other.

Part (a): What is the mean and standard deviation of the amount she spends on breakfast daily?

Part (b): What is the mean and standard deviation of the amount she spends on breakfast weekly (7 days)?

Work Space: