Namevalexis Tuckermath125 Unit 8 Submission Assignment Answer Formco ✓ Solved

NAME: Valexis Tucker MATH125: Unit 8 Submission Assignment Answer Form Counting Techniques ALL questions below must be answered. Show ALL step-by-step calculation . Upload this modified Answer Form to the intellipath Unit 8 Submission lesson. Make sure you submit your work in a modified MS Word document; handwritten work will not be accepted . If you need assistance, please contact your course instructor.

Part A: Combinations and Permutations 1. Differentiate between permutations and combinations. How are they different? What is the formula for each? (15 points total for Question 1) How are they different? (5 points) ? Permutation Formula (5 points) ?

Combination Formula (5 points) ? 2. Each state has a standard format for license plates that includes a set number of alphanumeric characters. For this assignment, you can insert a picture of your state’s non-personalized license plate or provide a sample of the format in text. (20 points total for Question 2) Your State’s Name (1 point) ? Picture of a License Plate from Your State (or a Sample) (1 point) ?

Describe the Rule for Your State’s Non-personalized License Plates (1 point) ? a. Determine the number of different license plates that can be created using this format. Assume that a license plate consists of seven alphanumeric characters using numbers (0–9) and capital letters (A–Z). Find how many unique license plates can be printed using all alphanumeric characters only once. Is this a permutation or combination?

Why? (2 points) ? What formula from Question 1 will you use to solve the problem? (1 point) ? Solution: (4 points) ? Show your work here: b. You and a friend are witnesses of a car accident in your state.

But you can only remember a few of the first alphanumeric characters on the license plate. How many alphanumeric characters do you remember? (1 point) ? (Select a number from 2 to 5) What are the characters at the beginning? (1 point) ? How many license plates start with these alphanumeric characters? (4 points) ? Show your work here: How many license plates have been eliminated? (4 points) ? Show your work: 3.

Your community has asked you to help the Young Men's Christian Association (YMCA) sports director organize a season of sports. You need to put together the teams. For the soccer teams, athletes signed up with three different age groups. How many different ways can you organize teams of 10 for each age group? (15 points total for Question 3) Are these a permutation or combination? Why? (2 points) ?

What formula from Question 1 will you use to solve the problem? (1 point) ? How many students signed up for soccer? (1 point) ? (Select a multiple of 10, from 30 to 100) How many kids signed up for Little Tykes under the age of seven? (1 point) ? (Select a multiple of 10, of at least 20) How many kids signed up for Big Kids between the ages 8 and 12? (1 point) ? (Select a multiple of 10, of at least 20) How many kids signed up for Teens between the ages 13 and 18? (1 point) ? (Select a multiple of 10, of at least 20) How many different ways can you create teams of 10 for the Little Tykes grade level? (2 points) ? Show your work here: (2 points) If age levels did not matter, how many different ways can you create teams of 10? (2 points) ?

Show your work here: (2 points) Part B: Probabilities and Odds 4. For this set of exercises, you will need one standard six-sided dice. If you do not have one, you can use virtual dice: ( 40 points total for Question 4) a. First, differentiate between odds and probability . How are odds and probability different? (2 points) ?

What is the odds in favor ratio? (3 points) ? What is the probability of an event ratio? (3 points) ? What are the odds of rolling a three? Simplify all fraction answers. (2 points) ? What is the theoretical probability of rolling a three?

Simplify all fraction answers. (2 points) ? b. Reflect on the previous question’s answer outcome. First, convert the fraction to a percent. Percent Probability Theoretical Probability (Rounded to the nearest whole percent.) (2 points) ? Next, given the likelihood scale table above, what term best describes your answer?

Likelihood Scale Term (2 points) ? c. What if someone challenged you to never roll a 3? If you were to roll the dice 18 times, what would be the empirical probability of never getting a three? Percent Probability Solution (Rounded to the Nearest Whole Percent) (2 points) ? Likelihood Scale Term (2 points) ?

Show your work here: (2 points) d. After 18 rolls, what would be the empirical probability of getting a three on at least one of those rolls? Also, list the likelihood scale term from the table above. Percent Probability Empirical Probability (Rounded to the Nearest Whole Percent) (2 points) ? Likelihood Scale Term (2 points) ?

Show your work: (2 points) What do you notice about the answers for parts c and d above? (2 points) ? e. Roll the dice 18 times and keep track of what is rolled in the table below. (2 points) Roll # Dice Roll # Dice Roll # Dice Roll 1 ? Roll 7 ? Roll 13 ? Roll 2 ?

Roll 8 ? Roll 14 ? Roll 3 ? Roll 9 ? Roll 15 ?

Roll 4 ? Roll 10 ? Roll 16 ? Roll 5 ? Roll 11 ?

Roll 17 ? Roll 6 ? Roll 12 ? Roll 18 ? f. Based on your dice rolls, what is the experimental probability of rolling a three, out of 18 rolls?

Also, list the likelihood scale term from the table above. Percent Probability Experimental Probability (Rounded to the Nearest Whole Percent) (2 points) ? Likelihood Scale Term (2 points) ? Show your work here: (2 points) With regard to the likelihood scale terms for each, how did this differ from both the theoretical and empirical probabilities? (2 points) ?

Paper for above instructions

Part A: Combinations and Permutations

1. Differentiate Between Permutations and Combinations.
Permutations and combinations are mathematical concepts used for counting and arranging items.
How are they different?
The primary difference lies in the order of arrangement:
- Permutations refer to the arrangements of items where the order does matter. For example, the arrangement of runners in a race (1st, 2nd, 3rd) varies based on their finishing positions.
- Combinations, on the other hand, refer to selections of items where the order does not matter. For example, choosing a committee from a group is a combination; the arrangement of the members is irrelevant.
Permutation Formula:
The formula for permutations is given by:
\[ P(n, r) = \frac{n!}{(n - r)!} \]
where \( n \) is the total number of items, \( r \) is the number of items to arrange, and \( n! \) is the factorial of \( n \).
Combination Formula:
The formula for combinations is:
\[ C(n, r) = \frac{n!}{r!(n - r)!} \]
where \( n \) is the total number of items, \( r \) is the number of items to select, and \( r! \) is the factorial of \( r \).
2. License Plate Format for [Your State].
Your State’s Name: [Insert your state name here, e.g., "California"]
Picture of a License Plate or Sample Format:
(Insert a picture or sketch format here - for example: "ABC 1234")
Rule Description for Non-Personalized License Plates:
In California, personalized license plates typically consist of a combination of 7 alphanumeric characters that can include both uppercase letters (A-Z) and digits (0-9), but cannot use the numeral '0' at the beginning.
a. Determine the Number of License Plates:
To create the number of different license plates, we analyze the total possibilities:
The total number of alphanumeric characters (0-9 and A-Z) = 10 (digits) + 26 (letters) = 36 characters.
The license plate consists of 7 characters, and since we can use each character only once, we are dealing with permutations.
Thus, the number of different license plates is calculated as:
\[
P(36, 7) = \frac{36!}{(36 - 7)!} = \frac{36!}{29!} = 36 \times 35 \times 34 \times 33 \times 32 \times 31 \times 30
\]
Solution Calculation:
Calculating that:
\[
= 36 \times 35 \times 34 \times 33 \times 32 \times 31 \times 30 = 9,831,032,960
\]
Thus, the total number of unique license plates is 9,831,032,960.
This is a permutation because the order of the characters matters.
What formula from Question 1 will you use to solve the problem?
I will use the permutation formula \( P(n, r) \).
b. Alphanumeric Characters Remembered:
- Number of Characters Remembered: 3
- Characters: Let's say, "ABC".
How many license plates start with these characters?
The remaining characters can be any of the other 33 characters (since "A", "B", and "C" are already used).
So, the number of license plates starting with "ABC" is given by:
\[
P(33, 4) = \frac{33!}{(33 - 4)!} = 33 \times 32 \times 31 \times 30 = 33,330,240
\]
How many license plates have been eliminated?
To find the total eliminated plates, we need to subtract the number of plates starting with “ABC” from the total plates.
Total license plates = 9,831,032,960
So the eliminated plates = \( 9,831,032,960 - 33,330,240 = 9,797,702,720 \).

3. Team Organization for YMCA Sports

How many different ways can you organize teams?
These scenarios involve choosing athletes for teams where the order does not matter, hence, they are combinations.
Number of Students Signed Up for Soccer: 60
Little Tykes Under 7: 30
Big Kids 8-12: 20
Teens 13-18: 20
To find the number of ways to organize teams of 10 for Little Tykes:
\[
C(30, 10) = \frac{30!}{10!(30 - 10)!}
\]
Calculating this:
\[
C(30, 10) = \frac{30 \times 29 \times 28 \times 27 \times 26 \times 25 \times 24 \times 23 \times 22 \times 21}{10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1} = 30045015
\]
Thus, there are 30,045,015 ways to organize Little Tykes.
If age levels do not matter, how many different ways can you create teams of 10?
Using 60 students, the total combinations are:
\[
C(60, 10) = \frac{60!}{10!(60 - 10)!}
\]
Calculating this gives:
\[
C(60, 10) = 75,394,027,566
\]

Part B: Probabilities and Odds

4. Differentiate Between Odds and Probability.
- Odds express the likelihood of an event compared to the likelihood of it not occurring. For instance, if the odds of rolling a 3 are 1-5, there is 1 favorable outcome and 5 unfavorable outcomes.
- Probability is the ratio of favorable outcomes to the total number of possible outcomes.
Odds in Favor Ratio:
For a standard die, the odds in favor of rolling a 3:
\[
\text{Odds} = 1 \text{ (favorable)} : 5 \text{ (unfavorable)}
\]
Probability of an Event Ratio:
The probability of rolling a 3 is calculated as:
\[
\text{Probability} = \frac{1}{6}
\]
Odds of Rolling a Three:
Simplified, that ratio is 1:5.
Theoretical Probability of Rolling a Three:
This is simply \( \frac{1 \text{ (favorable)}}{6 \text{ (total outcomes)}} = \frac{1}{6} \).
Converting Fraction to Percent:
To convert \( \frac{1}{6} \) to a percent:
\[
\frac{1}{6} \times 100 = 16.67\% \approx 17\%
\]
Likelihood Scale Term:
This would be "Rare."
Empirical Probability of Never Getting a Three in 18 Rolls:
If the probability of rolling a three is \( \frac{1}{6} \), the probability of not rolling a three is \( 1 - \frac{1}{6} = \frac{5}{6} \).
Using the formula for the probability of independent events:
\[
P(\text{no 3 in 18 rolls}) = \left(\frac{5}{6}\right)^{18} \approx 0.0207 \text{ or } 2\%
\]
Likelihood Scale Term:
This would be "Rare."

Conclusion

This assignment has detailed the differences between permutations and combinations and provided examples relating to license plates and team organization. Further, it analyzed the nuances of odds and probability related to dice rolls, culminating in a broad understanding of statistical counting and measurement fundamentals.

References

1. Bluman, A. G. (2018). Elementary Statistics: A Step by Step Approach. McGraw-Hill Education.
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5. Ross, S. M. (2019). Introduction to Probability and Statistics. Academic Press.
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8. Moore, D. S., & McCabe, G. P. (2015). Introduction to the Practice of Statistics. W.H. Freeman.
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