Q1. What is the probability that you get your first payout within the first 100 plays on a slot machine that pays out 1/1000th of the time?

Background

Topic: Geometric Probability Distribution

This question tests your understanding of the geometric distribution, which models the probability of the first success on the nth trial in a sequence of independent Bernoulli trials.

Key Terms and Formulas

• Geometric Distribution: Probability of first success on or before the nth trial.

• Probability of success on a single trial:

• Cumulative probability formula:

Step-by-Step Guidance

1. Identify the probability of success on a single play: .

2. Recognize that you want the probability of getting the first payout within 100 plays, i.e., .

3. Use the cumulative geometric probability formula: .

4. Plug in and into the formula, but do not compute the final value yet.

Try solving on your own before revealing the answer!

Q2. What are the odds of between 2000 and 2400 lightning strikes occurring in Sarasota in a year, given an average of 2,500 strikes per year?

Background

Topic: Poisson Distribution

This question tests your ability to use the Poisson distribution to find the probability of a certain number of events occurring in a fixed interval, given the average rate.

Key Terms and Formulas

• Poisson Distribution:

• Mean (expected number of events):

• "Between 2000 and 2400" means

Step-by-Step Guidance

1. Identify the mean number of events: .

2. Set up the probability you need: .

3. Recognize that for large , the Poisson can be approximated by a normal distribution with and .

4. Set up the normal approximation: , and find the -scores for and .

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Q3. Draw a tree diagram for the experiment “drawing 3 cards without replacement” from cards with values 3, 5, 6, and 9. How many outcomes are there?

Background

Topic: Counting Principles, Tree Diagrams, Permutations

This question tests your understanding of how to represent outcomes of sequential events (without replacement) using a tree diagram and how to count the total number of possible outcomes.

Key Terms and Formulas

• Tree Diagram: Visual representation of all possible outcomes.

• Permutation: Number of ways to arrange objects from without replacement:

Step-by-Step Guidance

1. List the four cards: 3, 5, 6, 9.

2. For the first draw, you have 4 choices; for the second, 3 choices; for the third, 2 choices.

3. Draw the first level of the tree with 4 branches (one for each card).

4. For each branch, draw the next level with 3 branches (remaining cards), and repeat for the third draw.

5. Set up the formula for the total number of outcomes: .

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Q4. For two events A and B, with , , and :

• (a) Are the events independent? Why or why not?

• (b) Calculate .

Background

Topic: Probability Rules – Independence, Addition Rule

This question tests your understanding of independence of events and how to use the addition rule for probabilities.

Key Terms and Formulas

• Independence: Events A and B are independent if

• Addition Rule:

Step-by-Step Guidance

1. For (a): Compute and compare it to to check independence.

2. For (b): Use the addition rule formula and plug in the given probabilities.

3. Set up the calculation for , but do not compute the final value yet.

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Q5. If you buy 100 boxes, each with a 20% chance to contain a hat, what are your odds of receiving 30 or more hats?

Background

Topic: Binomial Probability Distribution

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

Key Terms and Formulas

• Binomial Distribution:

• Probability of at least successes:

• Parameters: ,

Step-by-Step Guidance

1. Identify the parameters: , .

2. Set up the probability you need: .

3. Recognize that this is a cumulative binomial probability, which can be calculated as .

4. Set up the sum: .

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Q6. Find the mean and standard deviation of the random variable in the previous problem (number of hats in 100 boxes, each with a 20% chance of containing a hat).

Background

Topic: Mean and Standard Deviation of a Binomial Distribution

This question tests your ability to calculate the expected value (mean) and standard deviation for a binomial random variable.

Key Terms and Formulas

• Mean:

• Standard Deviation:

• Parameters: ,

Step-by-Step Guidance

1. Identify the parameters: , .

2. Plug these values into the mean formula: .

3. Plug these values into the standard deviation formula: .

4. Set up the calculations for both mean and standard deviation, but do not compute the final values yet.

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Q7. What percentage of a normal distribution with mean 135 and standard deviation 15 lies between 100 and 148?

Background

Topic: Normal Distribution, Z-scores, Probability

This question tests your ability to use the standard normal distribution to find the probability that a value falls within a certain range.

Key Terms and Formulas

• Z-score:

• Standard Normal Table: Used to find probabilities associated with Z-scores.

Step-by-Step Guidance

1. Identify the mean () and standard deviation ().

2. Calculate the Z-score for and using the formula .

3. Use the standard normal table to find the probabilities corresponding to these Z-scores.

4. Set up the calculation for the percentage between these two values, but do not compute the final value yet.

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Q8. Suppose you remove all Queens and Kings from a standard deck of 52 cards.

• (a) What is the probability that you don’t draw a 6 when drawing one card?

• (b) What is the probability that neither of two cards drawn without replacement is a 6?

Background

Topic: Probability with Modified Decks, Without Replacement

This question tests your ability to calculate probabilities with a modified deck and to handle dependent events (without replacement).

Key Terms and Formulas

• Number of cards after removing all Queens and Kings:

• Probability (single draw):

• Probability (two draws, without replacement):

Step-by-Step Guidance

1. For (a): Determine how many 6s remain in the deck and the total number of cards.

2. Calculate for one draw.

3. For (b): Set up the probability for two draws without replacement, using the multiplication rule.

4. Set up the calculation for the second draw, considering one less card and possibly one less 6.

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Q9. How many distinct 7-digit phone numbers are there that don’t contain any 0s and don’t repeat any digits?

Background

Topic: Counting, Permutations with Restrictions

This question tests your ability to count the number of arrangements (permutations) with restrictions (no zeros, no repeats).

Key Terms and Formulas

• Digits allowed: 1–9 (since 0 is not allowed)

• Number of ways to arrange 7 digits from 9 without repetition:

Step-by-Step Guidance

1. List the available digits: 1, 2, 3, 4, 5, 6, 7, 8, 9.

2. Recognize that you are choosing and arranging 7 digits from these 9, with no repeats.

3. Set up the permutation formula: .

4. Write out the expanded multiplication for .

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Q10. How many ways are there to order a meal consisting of 1 appetizer, no entrees, and 3 different desserts from 7 appetizers, 10 entrees, and 5 desserts?

Background

Topic: Counting, Combinations and Multiplication Principle

This question tests your ability to count the number of ways to make selections from different categories, using combinations and the multiplication principle.

Key Terms and Formulas

• Number of ways to choose 1 appetizer:

• Number of ways to choose 3 different desserts:

• Total ways: Multiply the number of ways for each category.

Step-by-Step Guidance

1. Set up the number of ways to choose 1 appetizer from 7: .

2. Set up the number of ways to choose 3 desserts from 5: .

3. Multiply the two results to get the total number of meal combinations.

4. Write out the expanded combinations: , .

Try solving on your own before revealing the answer!