# Module 4-5 Counting, Probability, and Expected Value

*Show all work to receive credit. Any written questions should be at least 3 sentences long.*

**Counting**

- A gift-wrapping store has 8 shapes of boxes, 14 types of wrapping paper, and 12 different bows. How many different gift-wrapping options are available at this store?
- Four friends go to the movies. How many different ways can they sit in a row?
- You are creating a 4-letter password using only the 26 lowercase letters from the alphabet. How many unique passwords could you create (assuming the password does not need to spell a real word) if:
- Repeating letters is allowed?

- Repeating letters are not allowed?

- What are some passwords that are possible in a. that are not possible in b.? What are some passwords that are possible in either scenario?

- Which of the following do I have to check before using the nPr? There may be more than one correct answer:

- Repeating is allowed
- Repeating is not allowed
- The order matters
- The order does not matter

Explain in a complete sentence why you chose the options for the previous question:

- Which of the following do I have to check before using nCr? There may be more than one correct answer:

- Repeating is allowed
- Repeating is not allowed
- The order matters
- The order does not matter

Explain in a complete sentence why you chose the options for the previous question:

- Two people out of a group of 75 will win tickets to an upcoming concert. How many different groups of two are possible?
- Barry is hosting a Super Bowl party and offers 7 different kinds of chip dip. If a party goer can choose any number of chip dips for their chips, how many chip dip combinations are possible?

- There are 30 students in the classroom competing for classroom prizes. Only the first two students whose name is drawn will win a prize. The first student wins 5 bonus points and the second student wins 2 bonus points, both on the upcoming test. If you are one of the 30 students, what is the probability you will win the top prize and your best friend wins the second prize?

**Probability and Odds**

Answer all questions with a fraction in lowest terms. If you’d like, you can also write them as a percent.

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**If you draw one card at random, what is the probability that card is a(n)…**

- Heart?

- 7 of diamonds?
- face card or a club?

__Given the card is a club__, what is the probability a card drawn at random will be a(n)…

- 8?
- 10 or ace?

**You are choosing two cards, without replacing the first card. What is the probability you choose…**

- a 7 then a 3?
- two consecutive fours?
- two consecutive diamonds?

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**You are choosing two cards, replacing the first card in the deck after it has been drawn. What is the probability you choose…**

- a 7 then a 3?
- two consecutive fours?
- two consecutive diamonds?

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- A card is drawn from a standard deck of 52 cards, and then replaced in the deck. Find the probability that at least one king is drawn by the fourth draw. Round your answer to two decimal places. Do not round until your final calculation.

- Bill and Sam are playing a game where they roll a die and draw a card from a deck. What is the probability of rolling a 5 and then drawing a 5 from the deck of cards? Write your answer as a fraction in lowest terms.

Use the formulas and to answer the following question:

- Suppose on a particular day, the probability (among the entire population) of getting into a car accident is 0.04, the probability of being a texter-and-driver is 0.14, and p(car accident
*or*being a texter-and-driver)=0.15. Find the probability a person was in a car accident*given*that they are a texter-and-driver. Is this higher or lower than the probability among the general population and why?

Remember the formula for expected value is: EV = Outcome1 * Probability1 + Outcome 2 * Probability2… and so on.

- You roll a die and win $1 if you get a 2 or 6, lose $2 if you get a 3, win $5 if you get a 6, and lose $3 if you get a 1. What is the expected value of this game? If you played the game repeatedly, would you expect to win money? Why?

- Suppose you are playing a game with two dice at a casino. If you roll double ones, you win $1, if you roll double twos, you win $2. This pattern continues, up to winning $6 for double sixes. If you roll anything besides doubles, you lose $1. What is the expected value of the game? Can you expect to win money by playing this game repeatedly? *Hint: Find the total number of outcomes of this game first. Only six outcomes win money. The other outcomes lose money. You’ll find the expected value with these outcomes. Good luck 😊