Probability Mastery Exam Test
PROBABILITY MASTERY
Comprehensive Unit Assessment
Name:
Date:
Part I: Probability Essentials
Solve the following 15 multiple-choice questions. Select the best answer for each.
- Which of the following values cannot represent the probability of an event?
A 0.71
B 1/8
C 120%
D 0.5
- In a random sample of 375 married couples, 34 couples shared all four personality preferences. What is the probability that a randomly chosen couple from this sample shares all four preferences?
A 0.091
B 0.150
C 0.250
D 0.375
- Plain M&Ms have the following color distribution: Orange (10%) and Blue (10%). What is the probability of selecting an orange or a blue candy?
A 1%
B 10%
C 20%
D 100%
- If the probability of selecting a brown M&M is 30%, what is the probability of selecting a candy that is not brown?
A 0.30
B 0.50
C 0.60
D 0.70
- In Arches National Park, 18 out of 288 arches are "75 feet and higher." What is the estimated probability that a randomly chosen arch is at least 75 feet tall?
A 0.0625
B 0.1800
C 0.2880
D 0.7500
- When rolling two fair dice (one green, one red), are the outcomes on the two dice considered independent?
A No, they are dependent.
B Yes, they are independent.
C Only if different colors.
D Only if rolled together.
- What is the probability of rolling a "5" on a green die and a "3" on a red die?
A 2/6
B 1/12
C
C 1/36
D 8/36
- What is the probability of getting a sum of 7 when rolling two fair dice?
A 1/6
B 1/7
C 7/36
D 5/36
- You draw two cards from a standard deck without replacing the first card. Are the outcomes on the two cards independent?
A Yes, cards don't affect each other.
B No, because the deck changes.
C Yes, if you shuffle afterwards.
D Only if different suits.
- Drawing two cards without replacement, what is the probability of drawing an Ace first and a King second?
A (4/52) × (4/52)
B (4/52) × (4/51)
C (4/52) + (4/51)
D 8/52
- If you flip a fair coin three times, how many total sequences (outcomes) are possible in the sample space?
A 3
B 6
C 8
D 9
- In the sample space of flipping a coin three times, how many of the sequences contain exactly two heads?
A 2
B 3
C 4
D 1
- An urn contains 6 identical balls: 2 red, 3 blue, and 1 yellow. What is the probability that the first ball drawn is blue?
A 1/6
B 1/3
C 1/2
D 3/5
- You flip a coin and then toss a six-sided die. What is the probability of getting a "Head" on the coin and a number greater than "4" on the die?
A 1/4
B 1/6
C 1/12
D 2/6
- If you guess randomly on three multiple-choice questions, each with four possible responses, what is the probability of getting all three answers correct?
A 1/12
B 1/64
C 3/4
D 1/16
Part II: Counting & Principles
Analyze the scenarios carefully and select the best answer for each question.
- A sales representative must visit five cities: Omaha, Dallas, Wichita, Oklahoma City, and Denver. Using the multiplication rule of counting, how many different orders are possible for the visit?
A 25
B 120
C 24
D 60
- Which of the following would be classified as a CONTINUOUS random variable?
A The number of books in a college bookstore
B The number of lightning strikes in a park on a given day
C The speed of an airplane during flight
D The number of goals scored in a soccer match
- Which of the following probability distributions is INVALID?
Option A
<table class="w-full text-center border-collapse text-sm"><tbody><tr class="border-b border-slate-200"><th class="p-1 bg-slate-50 border-r border-slate-200 font-bold">x</th><td class="p-1 border-r border-slate-200">0</td><td class="p-1 border-r border-slate-200">1</td><td class="p-1">2</td></tr><tr><th class="p-1 bg-slate-50 border-r border-slate-200 font-bold">P(x)</th><td class="p-1 border-r border-slate-200">0.25</td><td class="p-1 border-r border-slate-200">0.60</td><td class="p-1">0.15</td></tr></tbody></table>
Option B
<table class="w-full text-center border-collapse text-sm"><tbody><tr class="border-b border-slate-200"><th class="p-1 bg-slate-50 border-r border-slate-200 font-bold">x</th><td class="p-1 border-r border-slate-200">0</td><td class="p-1 border-r border-slate-200">1</td><td class="p-1">2</td></tr><tr><th class="p-1 bg-slate-50 border-r border-slate-200 font-bold">P(x)</th><td class="p-1 border-r border-slate-200">0.30</td><td class="p-1 border-r border-slate-200">0.50</td><td class="p-1">0.10</td></tr></tbody></table>
Option C
<table class="w-full text-center border-collapse text-sm"><tbody><tr class="border-b border-slate-200"><th class="p-1 bg-slate-50 border-r border-slate-200 font-bold">x</th><td class="p-1 border-r border-slate-200">0</td><td class="p-1 border-r border-slate-200">1</td><td class="p-1">2</td></tr><tr><th class="p-1 bg-slate-50 border-r border-slate-200 font-bold">P(x)</th><td class="p-1 border-r border-slate-200">0.10</td><td class="p-1 border-r border-slate-200">0.80</td><td class="p-1">0.10</td></tr></tbody></table>
Option D
<table class="w-full text-center border-collapse text-sm"><tbody><tr class="border-b border-slate-200"><th class="p-1 bg-slate-50 border-r border-slate-200 font-bold">x</th><td class="p-1 border-r border-slate-200">0</td><td class="p-1 border-r border-slate-200">1</td><td class="p-1">2</td></tr><tr><th class="p-1 bg-slate-50 border-r border-slate-200 font-bold">P(x)</th><td class="p-1 border-r border-slate-200">0.50</td><td class="p-1 border-r border-slate-200">0.25</td><td class="p-1">0.25</td></tr></tbody></table>
- Compute the value of \( P_{7,2} \).
A 14
B 21
C 42
D 49
- Compute the value of \( C_{8,3} \).
A 336
B 56
C 24
D 40,320
- There are three distinct nursing positions to be filled: Day Supervisor, Night Supervisor, and Nursing Coordinator. If there are 10 qualified candidates, how many ways can these positions be filled?
A 120
B 1,000
C 720
D 30
- A deli special lunch offers a choice of 3 sandwiches, 4 salads, and 5 desserts. How many different lunches can be ordered if each consists of one sandwich, one salad, and one dessert?
A 12
B 60
C 15
D 23
- When tossing a pair of dice, how many outcomes result in both dice showing an even number?
A 9
B 6
C 12
D 36
- You draw one card from each of two separate decks (52 cards each). What is the probability of drawing two Kings? (Recall: 4 Kings per deck).
A \( \frac{1}{169} \)
B \( \frac{8}{52} \)
C \( \frac{16}{2704} \)
D Both A and C are correct
- In a lottery game, there are 10 finalists. Three grand prize winners are chosen. If the order of selection does not matter, how many different groups of winners are possible?
A 720
B 120
C 30
D 1,000
Part III: Binomial Distribution
Solve each problem below. You must show your step-by-step solutions to receive full credit.
Reference: Binomial Probability Formula
\[ P(r) = \frac{n!}{r!(n-r)!} p^r q^{n-r} \]
- A professional soccer player has a 75% success rate on penalty kicks. What is the probability that the player makes exactly four out of five penalty kicks?
Show your work here
- A manufacturing plant produces light bulbs with a 5% defect rate. If a random sample of ten bulbs is selected, what is the probability that exactly one bulb is defective?
Show your work here
- A student takes a multiple-choice quiz with 10 questions. Each question has four possible options, and only one is correct. If the student guesses randomly on every question, what is the probability they get exactly three questions correct?
Show your work here
- A local fisherman has a 20% chance of catching a fish on any given cast. If he makes eight casts, what is the probability that he catches exactly two fish?
Show your work here
- The weather forecast predicts a 30% chance of rain for each day of a 5-day vacation. What is the probability that it rains on exactly two of those days?
Show your work here
Probability Mastery Exam Answer Key
ANSWER KEY
Probability Mastery Exam (30 Questions)
TEACHER REFERENCE
Part I: Probability Essentials
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C (120%)
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A (0.091)
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C (20%)
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D (0.70)
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A (0.0625)
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B (Yes, independent)
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C (1/36)
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A (1/6)
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B (No, deck changes)
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B (4/52 x 4/51)
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C (8)
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B (3)
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C (1/2)
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B (1/6)
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B (1/64)
Part II: Counting & Principles
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B (120)
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C (Airplane speed)
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B (Sums to 0.9)
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C (42)
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B (56)
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C (720)
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B (60)
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A (9)
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D (Both A and C)
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B (120)
Part III: Binomial Distribution (Step-by-Step)
26. Success rate 75%, exactly 4/5:
\[ P(4) = \binom{5}{4} (0.75)^4 (0.25)^1 \] \[ P(4) = 5 \times 0.3164 \times 0.25 \approx \mathbf{0.3955} \]
27. Defect rate 5%, n=10, exactly 1:
\[ P(1) = \binom{10}{1} (0.05)^1 (0.95)^9 \] \[ P(1) = 10 \times 0.05 \times 0.6302 \approx \mathbf{0.3151} \]
28. 10 questions, 1/4 chance, guess 3 correct:
\[ P(3) = \binom{10}{3} (0.25)^3 (0.75)^7 \] \[ P(3) = 120 \times 0.0156 \times 0.1335 \approx \mathbf{0.2503} \]
29. 20% chance, 8 casts, exactly 2 caught:
\[ P(2) = \binom{8}{2} (0.20)^2 (0.80)^6 \] \[ P(2) = 28 \times 0.04 \times 0.2621 \approx \mathbf{0.2936} \]
30. 30% rain, 5 days, exactly 2 rains:
\[ P(2) = \binom{5}{2} (0.30)^2 (0.70)^3 \] \[ P(2) = 10 \times 0.09 \times 0.343 \approx \mathbf{0.3087} \]
