Probability and Statistics Practice Test 2 – Guided Study Notes

Q1. State which of the following are valid probabilities (and why): (a) -0.25 (b) 1.23 (c) 0.32% (d) 4 (e) 27%

Background

Topic: Probability Basics

This question tests your understanding of the definition of probability and the valid range for probability values.

Key Terms and Concepts:

• Probability: A measure of how likely an event is to occur, expressed as a number between 0 and 1 (inclusive), or equivalently, between 0% and 100%.

• Valid Probability: For any event, $0 \leq P(A) \leq 1$.

Step-by-Step Guidance

1. Check if each value is between 0 and 1 (inclusive). If a value is negative or greater than 1, it is not a valid probability.

2. For percentages, convert them to decimals by dividing by 100 before checking the range.

3. Explain why each value is or is not a valid probability based on the definition above.

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Q2. If a regular six-sided die is rolled, let Event A be: rolling an odd number. (a) List all outcomes in Event A. (b) List all outcomes in the complement of Event A. (c) List the entire sample space of the experiment.

Background

Topic: Sample Spaces and Events

This question tests your ability to identify outcomes for events and their complements, and to describe the sample space for a simple random experiment.

Key Terms and Concepts:

• Sample Space ($S$): The set of all possible outcomes of an experiment.

• Event: A subset of the sample space (e.g., rolling an odd number).

• Complement ($A'$): All outcomes in the sample space that are not in event $A$.

Step-by-Step Guidance

1. List all possible outcomes when rolling a six-sided die (the sample space).

2. Identify which outcomes are considered 'odd' numbers for Event A.

3. List the outcomes that are not odd (these make up the complement, $A'$).

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Q3. Are the following dependent or independent events (why or why not)? Event A: Roll a 6 on my first roll of a die. Event B: Roll a 6 on the second roll of a die.

Background

Topic: Independence of Events

This question tests your understanding of the difference between independent and dependent events in probability.

Key Terms and Concepts:

• Independent Events: Two events are independent if the occurrence of one does not affect the probability of the other.

• Dependent Events: Two events are dependent if the occurrence of one affects the probability of the other.

Step-by-Step Guidance

1. Consider whether the outcome of the first roll affects the outcome of the second roll when rolling a fair die.

2. Recall the definition of independence and apply it to this scenario.

3. Explain your reasoning based on the definitions above.

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Q4. Are the following events mutually exclusive or not (why or why not)? Event A: Score a 97 on this test. Event B: Score less than 80 on this test.

Background

Topic: Mutually Exclusive Events

This question tests your understanding of mutually exclusive (disjoint) events.

Key Terms and Concepts:

• Mutually Exclusive Events: Two events are mutually exclusive if they cannot both occur at the same time.

Step-by-Step Guidance

1. Consider whether it is possible for a student to both score a 97 and score less than 80 on the same test.

2. Apply the definition of mutually exclusive events to this scenario.

3. Explain your reasoning.

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Q5. For each random variable below, state whether it is discrete or continuous (and why): (a) Number of dogs in a dog park at a given time (b) Amount of time a dog spends in a dog park in a day (c) Number of students who answer this question correctly

Background

Topic: Types of Random Variables

This question tests your ability to distinguish between discrete and continuous random variables.

Key Terms and Concepts:

• Discrete Random Variable: Takes on countable values (often integers).

• Continuous Random Variable: Takes on any value within a given range (often measurements).

Step-by-Step Guidance

1. For each variable, ask: Can the values be counted (discrete) or measured on a continuous scale (continuous)?

2. Justify your classification for each variable.

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Q6. State whether or not the following table is a valid probability distribution (and why or why not):

X	P(x)
0	0.005
1	0.352
2	0.530
3	0.401
4	0.012

Background

Topic: Probability Distributions

This question tests your ability to recognize the requirements for a valid probability distribution.

Key Terms and Concepts:

• Probability Distribution: A table or function that gives the probabilities of all possible values of a random variable.

• Requirements: (1) Each probability $P(x)$ must be between 0 and 1, inclusive. (2) The sum of all probabilities must be 1.

Step-by-Step Guidance

1. Check that each $P(x)$ is between 0 and 1.

2. Add all the probabilities together to see if they sum to 1.

3. If either requirement is not met, explain why the table is not a valid probability distribution.

Try solving on your own before revealing the answer!