Q1. How many different sandwiches are possible given 3 types of bread, 2 types of cheese, 4 types of meat, and 3 sauces?

Background

Topic: Counting Principle (Multiplication Rule)

This question tests your understanding of how to count the total number of possible outcomes when making independent choices for each component of a sandwich.

Key Terms and Formulas

• Multiplication Rule: If there are ways to make the first choice, ways to make the second, ..., ways to make the k-th choice, then the total number of combinations is .

Step-by-Step Guidance

1. Identify the number of choices for each sandwich component: bread (3), cheese (2), meat (4), sauce (3).

2. Apply the multiplication rule: Multiply the number of choices for each component together.

3. Set up the expression: .

Try solving on your own before revealing the answer!

Q2. How many ways are there to select 6 students from 16 available students?

Background

Topic: Combinations

This question tests your ability to count the number of ways to choose a subset of items (students) from a larger set, where order does not matter.

Key Terms and Formulas

• Combination Formula:

• = total number of items, = number of items to choose

Step-by-Step Guidance

1. Identify (total students) and (students to select).

2. Write the combination formula:

3. Set up the factorials for calculation: is the product of all integers from 16 down to 1, is from 6 to 1, and is from 10 to 1.

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Q3. How many different routes are possible for a driver to make deliveries at 3 locations among 9 remaining locations?

Background

Topic: Permutations

This question tests your understanding of how to count the number of ways to arrange a subset of items (locations) where order matters.

Key Terms and Formulas

• Permutation Formula:

• = total items, = number to arrange

Step-by-Step Guidance

1. Identify (locations), (deliveries to make).

2. Write the permutation formula:

3. Set up the factorials: is the product of 9 down to 1, is from 6 to 1.

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Q4. What is the probability of correctly guessing a 3-digit PIN code on a 7-key keypad, with repetition allowed?

Background

Topic: Probability with Counting

This question tests your ability to calculate the probability of a specific outcome when all outcomes are equally likely and repetition is allowed.

Key Terms and Formulas

• Probability Formula:

• With repetition, total outcomes =

Step-by-Step Guidance

1. Determine the total number of possible PIN codes: (since each digit can be any of 7 keys).

2. There is only 1 correct PIN code (favorable outcome).

3. Set up the probability:

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Q5a. In how many ways can you select 4 colleges to apply to from 12 visited colleges?

Background

Topic: Combinations

This question tests your ability to count the number of ways to choose a subset where order does not matter.

Key Terms and Formulas

• Combination Formula:

Step-by-Step Guidance

1. Identify (colleges), (to select).

2. Set up the formula:

3. Prepare to compute the factorials for the calculation.

Try solving on your own before revealing the answer!

Q5b. In how many ways can you rank your top five colleges out of 12 visited colleges?

Background

Topic: Permutations

This question tests your ability to count the number of ways to arrange a subset where order matters.

Key Terms and Formulas

• Permutation Formula:

Step-by-Step Guidance

1. Identify (colleges), (to rank).

2. Set up the formula:

3. Prepare to compute the factorials for the calculation.

Try solving on your own before revealing the answer!

Q6a. Is the following a probability distribution? If so, find its mean and standard deviation.

Background

Topic: Probability Distributions, Mean and Standard Deviation

This question tests your ability to recognize a valid probability distribution and calculate its mean and standard deviation.

Key Terms and Formulas

• Probability Distribution: A table where all probabilities are between 0 and 1 and sum to 1.

• Mean (Expected Value):

• Standard Deviation:

Step-by-Step Guidance

1. Check that all values are between 0 and 1 and sum to 1.

2. If so, it's a valid probability distribution.

3. Calculate the mean: by multiplying each by its and summing.

4. Calculate the standard deviation: .

Try solving on your own before revealing the answer!

Q6b. Is the following a probability distribution? If not, why not?

Background

Topic: Probability Distributions

This question tests your ability to recognize whether a table of probabilities is a valid probability distribution.

Key Terms and Formulas

• Probability Distribution: All probabilities must be between 0 and 1 and sum to 1.

Step-by-Step Guidance

1. Check that all values are between 0 and 1.

2. Add up all the values to see if they sum to 1.

3. If the sum is not 1, explain why this is not a valid probability distribution.

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Q7. In a random sample of 30 adults, what is the probability that at most 5 believe it is bad luck to walk under a ladder, given 12% of all adults believe this?

Background

Topic: Binomial Probability

This question tests your ability to use the binomial distribution to find the probability of a certain number of successes in a fixed number of trials.

Key Terms and Formulas

• Binomial Distribution:

• Cumulative Probability: is the sum of probabilities for up to .

• , ,

Step-by-Step Guidance

1. Recognize this is a binomial probability problem: , .

2. Set up the cumulative probability: .

3. Use the binomial cumulative distribution function (binomcdf) or sum the individual probabilities from to .

Try solving on your own before revealing the answer!

Q8. Does surveying 100 random college students and recording Yes/No to a question result in a binomial distribution?

Background

Topic: Binomial Distribution Criteria

This question tests your understanding of the requirements for a binomial distribution.

Key Terms and Formulas

• Binomial Distribution: Requires a fixed number of independent trials, each with two possible outcomes, and the probability of success is the same for each trial.

Step-by-Step Guidance

1. Check if the experiment has a fixed number of trials (100 students).

2. Check if each trial has two outcomes (Yes/No).

3. Determine if the variable of interest is the number of successes (e.g., number of Yes responses).

4. If only recording Yes/No but not counting the number of Yes responses, explain why this is not binomial.

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Q9a. What is the probability that exactly 15 out of 20 randomly selected adult smartphone users use their phones in meetings or classes, given 44% do so?

Background

Topic: Binomial Probability

This question tests your ability to use the binomial probability formula to find the probability of a specific number of successes.

Key Terms and Formulas

• Binomial Probability Formula:

• , ,

Step-by-Step Guidance

1. Identify the parameters: , , .

2. Set up the binomial probability formula: .

3. Calculate (the number of ways to choose 15 from 20).

4. Raise to the 15th power and to the 5th power.

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Q9b. What is the probability that at most 10 out of 20 randomly selected adult smartphone users use their phones in meetings or classes, given 44% do so?

Background

Topic: Binomial Cumulative Probability

This question tests your ability to use the binomial cumulative distribution function to find the probability of up to a certain number of successes.

Key Terms and Formulas

• Cumulative Binomial Probability:

• , ,

Step-by-Step Guidance

1. Identify the parameters: , , .

2. Set up the cumulative probability: .

3. Use the binomial cumulative distribution function (binomcdf) or sum the probabilities for to .

Try solving on your own before revealing the answer!