STA 2023 Test 2 Practice – Step-by-Step Statistics Guidance

Q1. Identify the sample space: Grade level of a public school student

Background

Topic: Sample Space in Probability

This question is about identifying all possible outcomes (the sample space) for a random experiment—in this case, the possible grade levels a public school student could be in.

Key Terms:

• Sample Space (S): The set of all possible outcomes of a random experiment.

Step-by-Step Guidance

1. Think about the range of grade levels typically found in public schools (e.g., Kindergarten through 12th grade).

2. List each grade level as a separate outcome in your sample space.

3. Write your sample space as a set, using curly braces { } and separating each outcome with a comma.

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Q2. Draw a tree diagram and identify the sample space for the answers to a three-question true/false test.

Background

Topic: Counting Outcomes and Tree Diagrams

This question tests your ability to use a tree diagram to visualize all possible sequences of answers (True or False) for a three-question test.

Key Terms:

• Tree Diagram: A graphical representation showing all possible outcomes of a sequence of events.

• Sample Space: The set of all possible answer combinations (e.g., TTF, FTT, etc.).

Step-by-Step Guidance

1. Start by drawing a branch for each possible answer to the first question (True or False).

2. From each of those branches, draw two more branches for the second question (True or False).

3. Repeat for the third question, branching from each previous outcome.

4. List all the possible sequences (e.g., TTT, TTF, TFT, TFF, etc.) as your sample space.

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Q3. Determine the number of outcomes in the following event: A computer chooses a number between 1 and 1000. The number is even.

Background

Topic: Counting Outcomes

This question is about determining how many even numbers are in a given range.

Key Terms and Formulas:

• Even Number: An integer divisible by 2.

• Counting Even Numbers: Use the formula for the number of terms in an arithmetic sequence.

Step-by-Step Guidance

1. Identify the smallest and largest even numbers in the range 1 to 1000.

2. Set up the arithmetic sequence: first term $a_1 = 2$, last term $a_n = 1000$, common difference $d = 2$.

3. Use the formula for the nth term: $a_n = a_1 + (n-1)d$ and solve for $n$.

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Q4. The new McDonald’s meal deal allows you to select from 3 sides, 8 entrees, and 4 desserts. How many unique meals can be formed?

Background

Topic: Counting Principle (Multiplication Rule)

This question tests your understanding of how to count the total number of possible combinations when making independent choices.

Key Formula:

• Multiplication Rule: If there are $m$ ways to do one thing, $n$ ways to do another, and $p$ ways to do a third, then there are $m \times n \times p$ ways to do all three.

Step-by-Step Guidance

1. Identify the number of choices for each category: sides (3), entrees (8), desserts (4).

2. Multiply the number of choices together: $3 \times 8 \times 4$.

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Q5. Find the probability of rolling a six-sided die and getting an even number.

Background

Topic: Basic Probability

This question is about finding the probability of a specific event (rolling an even number) when all outcomes are equally likely.

Key Terms and Formula:

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

Step-by-Step Guidance

1. List all possible outcomes when rolling a six-sided die: 1, 2, 3, 4, 5, 6.

2. Identify which outcomes are even numbers.

3. Count the number of even outcomes and the total number of outcomes.

4. Set up the probability formula using these counts.

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Q6. There is an 80% chance that a student will remember to bring a formula sheet to class. What is the probability that the student will forget?

Background

Topic: Complement Rule in Probability

This question tests your understanding of complementary events (the probability of an event not happening).

Key Formula:

• Complement Rule: $P(A^c) = 1 - P(A)$, where $A^c$ is the complement of event $A$.

Step-by-Step Guidance

1. Let $A$ be the event that the student remembers the formula sheet. $P(A) = 0.80$.

2. The event that the student forgets is the complement, $A^c$.

3. Use the complement rule: $P(A^c) = 1 - P(A)$.

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Q7. A Lighthouse ID consists of 6 digits past the A00 start. What is the probability that you randomly guess my Lighthouse ID?

Background

Topic: Probability of a Specific Outcome

This question is about finding the probability of guessing a specific sequence of digits.

Key Terms and Formula:

• Probability of a Specific Sequence: $P = \frac{1}{\text{Total number of possible sequences}}$

Step-by-Step Guidance

1. Determine how many possible digits can be used for each of the 6 positions (usually 0-9, so 10 choices each).

2. Calculate the total number of possible 6-digit sequences: $10^6$.

3. Set up the probability as $P = \frac{1}{10^6}$.

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Q8. Determine the number of license plates if the plate consists of 6 characters—the first three of which are letters and the last three of which are numbers, with the first letter not an O and the last digit not a 0.

Background

Topic: Counting with Restrictions

This question tests your ability to count the number of possible arrangements when there are restrictions on some positions.

Key Terms and Formula:

• Multiplication Rule: Multiply the number of choices for each position.

• Restrictions: First letter cannot be 'O'; last digit cannot be 0.

Step-by-Step Guidance

1. Count the number of possible letters for the first position (exclude 'O').

2. Count the number of possible letters for the second and third positions (all 26 letters).

3. Count the number of possible digits for the fourth and fifth positions (0-9, so 10 choices each).

4. Count the number of possible digits for the sixth position (1-9, since 0 is not allowed).

5. Multiply the number of choices for all six positions together.

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Q9. What is the probability that the license plate in #8 ends in an even number?

Background

Topic: Probability with Restrictions

This question builds on the previous one, asking for the probability that the last digit is even, given the restrictions.

Key Terms and Formula:

• Even Digits: 2, 4, 6, 8 (since 0 is not allowed).

• Probability: $P = \frac{\text{Number of favorable plates}}{\text{Total number of plates}}$

Step-by-Step Guidance

1. From your answer to #8, recall the total number of possible plates.

2. Determine how many choices for the last digit are even (from 1-9, only 2, 4, 6, 8).

3. Calculate the number of plates that end with an even digit by multiplying the number of choices for each position, using only the even options for the last digit.

4. Set up the probability as the ratio of favorable plates to total plates.

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Q10. What is the probability that the license plate in #8 starts with a letter in my last name (Pridgen)?

Background

Topic: Probability with Specific Letter Choices

This question asks for the probability that the first letter is one of the letters in 'Pridgen', given the restrictions from #8.

Key Terms and Formula:

• Letters in 'Pridgen': P, R, I, D, G, E, N (7 letters).

• Probability: $P = \frac{\text{Number of favorable plates}}{\text{Total number of plates}}$

Step-by-Step Guidance

1. Count the number of possible choices for the first letter (from the 7 letters in 'Pridgen', making sure 'O' is not included).

2. For the other positions, use the same counts as in #8.

3. Calculate the number of plates that start with a letter from 'Pridgen'.

4. Set up the probability as the ratio of favorable plates to total plates.

Try solving on your own before revealing the answer!