Binomial Battles Assessment Probability & Logic Assessment
Counting • Biconditionals • Binomials
Student Name
Date
Part I: Counting Principles
1. In a group of 12 nurses, 4 are chosen for a special night shift. Calculate the number of possible groups. Show your formula setup.
2. Calculate \( _{8}P_{3} \) for arranging 3 trophies on a shelf from a set of 8.
3. Calculate \( _{11}C_{5} \) for selecting 5 households out of 11.
4. A 5-digit security code uses unique digits. Is this a permutation or combination? Explain your reasoning.
5. How many ways can 7 friends be seated in a row at a theater? Show the calculation.
Part II: Logical Foundations
The biconditional statement is \( P \leftrightarrow Q \) (P if and only if Q).
6. Write the biconditional statement for: "If a distribution is valid, then its sum is 1" and "If the sum is 1, then the distribution is valid."
7. In logic, if \( P \leftrightarrow Q \) is true, what must be true about the truth values of P and Q?
8. Complete the truth table row: If P is False and Q is False, what is the value of \( P \leftrightarrow Q \)?
9. Rewrite as a biconditional: "A random variable is discrete provided that it has a finite outcomes, and vice versa."
10. Explain why "p is success if and only if q is failure" is a biconditional statement in probability.
Part III: Random Variables & Distributions
11. height of a nurse D / C: ________
12. number of room calls D / C: ________
Data Set A: Super Shopper Incomes
<table class="w-full text-center text-[10px] border-collapse bg-white"><tbody><tr class="bg-blue-50"><th class="border border-slate-300 p-1 text-blue-900">Midpoint x</th><td class="border border-slate-300 p-1">10</td><td class="border border-slate-300 p-1">20</td><td class="border border-slate-300 p-1">30</td><td class="border border-slate-300 p-1">40</td><td class="border border-slate-300 p-1">50</td><td class="border border-slate-300 p-1">60</td></tr><tr><th class="border border-slate-300 p-1 text-blue-900">P(x)</th><td class="border border-slate-300 p-1">0.21</td><td class="border border-slate-300 p-1">0.14</td><td class="border border-slate-300 p-1">0.22</td><td class="border border-slate-300 p-1">0.15</td><td class="border border-slate-300 p-1">0.20</td><td class="border border-slate-300 p-1">0.08</td></tr></tbody></table>
13. Calculate the expected income \( \mu \) for a super shopper. Show the summation setup.
14. Find \( P(x > 40) \) from Data Set A.
15. What is \( P(20 \le x \le 40) \)?
Data Set B: Family Size (%)
<table class="w-full text-center text-[10px] border-collapse bg-white"><tbody><tr class="bg-blue-50"><th class="border border-slate-300 p-1">Size x</th><td class="border border-slate-300 p-1">2</td><td class="border border-slate-300 p-1">3</td><td class="border border-slate-300 p-1">4</td><td class="border border-slate-300 p-1">5</td><td class="border border-slate-300 p-1">6</td><td class="border border-slate-300 p-1">7</td></tr><tr><th class="border border-slate-300 p-1">Percent</th><td class="border border-slate-300 p-1">42%</td><td class="border border-slate-300 p-1">23%</td><td class="border border-slate-300 p-1">21%</td><td class="border border-slate-300 p-1">10%</td><td class="border border-slate-300 p-1">3%</td><td class="border border-slate-300 p-1">1%</td></tr></tbody></table>
16. What is the probability a selected family has more than 5 members?
17. Calculate the expected family size \( \mu \). Round to 2 decimals.
18. If \( \sum x^2 P(x) = 11.80 \), find \( \sigma^2 \).
19. If \( \sigma^2 = 3.24 \), find \( \sigma \).
20. What is the probability that a family has fewer than 3 members?
Part IV: Binomial Battles
21. List the four requirements for a Binomial Experiment.
Scenario: Parole Success (p=0.25, n=5)
22. Probability exactly 0 return to prison. Show formula.
23. Probability at least 1 returns using the Complement Rule.
24. Quiz: n=10, p=0.20. Find probability of exactly 2 correct.
25. Marriage: n=10, p=0.80. Find probability of exactly 8 say yes.
26. If \( n=20, p=0.10 \), find \( \mu \).
27. If \( n=20, p=0.10 \), find \( \sigma \).
28. Ethics: n=7, p=0.10. Probability \( P(r \le 1) \).
29. Revenue: n=5, p=0.85. Probability exactly 4 "high" days.
30. Define \( q \) and its mathematical relationship to \( p \).
Binomial Battles Answer Key Answer Key
Binomial Battles Assessment
Part I: Counting
1. \( _{12}C_{4} = 495 \). Setup: \( \frac{12!}{4!8!} \)
2. \( _{8}P_{3} = 336 \)
3. \( _{11}C_{5} = 462 \)
4. Permutation. Order matters for unique security codes.
5. \( 7! = 5,040 \)
Part II: Logical Foundations
6. A probability distribution is valid if and only if its sum is equal to 1.
7. They must have the same truth value (both True or both False).
8. True.
9. A random variable is discrete if and only if it has a finite number of outcomes.
10. It is biconditional because if \( p \) is the success probability, then \( q \) must be the failure probability, and vice versa (\( p+q=1 \)).
Part III: Random Variables
11. Continuous
12. Discrete
13. \( \mu = 31.3 \)
14. \( 0.28 \) (\( 0.20 + 0.08 \))
15. \( 0.51 \) (\( 0.14 + 0.22 + 0.15 \))
16. \( 0.04 \) (\( 3\% + 1\% \))
17. \( \mu = 3.14 \)
18. \( \sigma^2 = 11.80 - (3.14)^2 = 1.9404 \)
19. \( \sigma = 1.8 \)
20. \( 0.42 \)
Part IV: Binomial Battles
21. 1) Fixed n; 2) Independent trials; 3) 2 Outcomes; 4) Constant p.
22. \( P(0) = (0.75)^5 = 0.2373 \)
23. \( 1 - 0.2373 = 0.7627 \)
24. \( _{10}C_{2} (0.2)^2 (0.8)^8 = 0.3020 \)
25. \( _{10}C_{8} (0.8)^8 (0.2)^2 = 0.3020 \)
26. \( \mu = 2.0 \)
27. \( \sigma = 1.3416 \)
28. \( P(0)+P(1) = 0.4783 + 0.3720 = 0.8503 \)
29. \( _{5}C_{4} (0.85)^4 (0.15)^1 = 0.3915 \)
30. \( q \) is the probability of failure; \( q = 1 - p \).
Binomial Battles MC Assessment Binomial Battles Assessment
Multiple Choice Format
Student Name
Date
Instructions
Select the best answer for each of the 30 questions. Circle your choice clearly. Round probability calculations to 4 decimal places where applicable.
Part I: Counting Principles
1. In a group of 12 nurses, 4 are chosen for a night shift. Which formula represents the number of ways to do this?
(A) 12P4 (11,880)
(B) 12 x 4 (48)
(C) 12C4 (495)
(D) 12^4 (20,736)
2. How many ways can 3 trophies be arranged on a shelf from a set of 8 distinct trophies?
(A) 56
(B) 336
(C) 512
(D) 24
3. A survey requires choosing 5 households from a list of 11. How many different groups are possible?
(A) 462
(B) 55,440
(C) 161,051
(D) 55
4. Which describes why a 5-digit security code (unique digits) is a permutation?
(A) Because the digits are chosen from a larger set.
(B) Because the order of digits changes the identity of the code.
(C) Because digits can be repeated in most codes.
(D) Because there are exactly 10 possible digits (0-9).
5. In how many distinct ways can 7 friends be seated in a row at a theater?
(A) 49
(B) 2,520
(C) 823,543
(D) 5,040
Part II: Logical Foundations
6. Which correctly represents the biconditional connection between "valid" and "sum is 1"?
(A) If a distribution is valid, then its sum is 1.
(B) A distribution is valid if and only if its sum is 1.
(C) If the sum is 1, then the distribution is valid.
(D) A distribution is valid only if the sum is not 1.
7. If the statement P ↔ Q is True, what must be true about P and Q?
(A) P must be True and Q must be False.
(B) P must be True.
(C) P and Q must have identical truth values.
(D) Either P is True or Q is True, but not both.
8. What is the truth value of P ↔ Q when P is False and Q is False?
(A) True
(B) False
(C) Undefined
(D) Impossible
9. Rewrite of: "Discrete variables have finite outcomes, and vice-versa"?
(A) If a variable has finite outcomes, it is discrete.
(B) Discrete variables have many outcomes.
(C) A variable is discrete if and only if it has finite outcomes.
Binomial Battles MC Key Answer Key
Binomial Battles MC Assessment
1. C Combination formula
2. B Permutation \( 8 \times 7 \times 6 \)
3. A Combination \( _{11}C_{5} \)
4. B Order changes identity
5. D \( 7! = 5,040 \)
6. B Biconditional phrasing
7. C Truth values match
8. A \( F \leftrightarrow F = T \)
9. C Standard logic phrasing
10. B Complement rule logic
11. B Measured quantity
12. A Counted quantity
13. A \( \mu = 31.3 \)
14. C \( 0.20 + 0.08 \)
15. B \( 0.14 + 0.22 + 0.15 \)
16. B \( 3\% + 1\% \)
17. B \( \mu = 3.14 \)
18. A \( 11.80 - 3.14^2 \)
19. C \( \sqrt{3.24} = 1.8 \)
20. A Only family size 2
21. C Variable must be discrete
22. A \( 0.75^5 = 0.2373 \)
23. B \( 1 - 0.2373 \)
24. C Binomial \( P(2) \)
25. A Binomial \( P(8) \)
26. B \( 20 \times 0.10 \)
27. C \( \sqrt{20 \times 0.1 \times 0.9} \)
28. B \( P(0) + P(1) \)
29. A \( _{5}C_{4} (0.85^4)(0.15) \)
30. C \( 1 - 0.37 \)
