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Exam 2 Review Math 1342

General Information

• Date: ________________

• Name: ________________________________

• Format: Multiple Choice (This exam will focus on applying concepts learned throughout the course, emphasizing not just correct answers but also understanding the underlying principles.)

Probability Concepts

1. Tree Diagrams

• Tree diagrams are a versatile tool used to visualize outcomes in complex probability scenarios, helping in breaking down events into manageable parts.

• Example with sports teams:   - Event: "They lose at least one game"
    - Possible Outcomes:
      - A) {WWW (Win all games), WWL (Win first two, lose third), WLW (Win first, lose second, win third), LWW (Lose first, win last two)}
      - B) {WWL, WLW, WLL, LWW, LWL, LLW, LLL}
      - C) {WWW, WWL, WLW, WLL, LWW, LWL, LLW}
      - D) {WWL, WLW, LWW}   - These outcomes help in calculating probabilities of different combinations occurring.

2. Basic Probability Calculations

• Flipping a Coin: When flipping a coin three times, the outcomes include:   - HHH, HHT, HTH, HTT, THH, THT, TTH, TTT.   - Probability of at least one head:
      - A) $rac{3}{4}$ (Incorrect)
      - B) $rac{7}{8}$ (Correct)
      - C) $rac{1}{2}$ (Incorrect)
      - D) $rac{1}{4}$ (Incorrect)
    - Understanding the complementary event helps in determining this probability effectively.

3. Rolling Dice

• Rolling two balanced dice results in 36 total possible outcomes, represented as pairs ranging from (1,1) to (6,6).

• For the event that the sum is 10:   - Possible pairs: (4, 6), (5, 5), (6, 4)
    - Total outcomes for this event:
      - A) $rac{1}{18}$
      - B) $rac{1}{9}$
      - C) $rac{5}{36}$ (Correct)
      - D) $rac{1}{12}$   - This also illustrates how specific events can be calculated using favorable outcomes over total outcomes.

4. Group Representation

• In a class, we have a gender distribution breakdown:   - Boys from Wilmette: 8
    - Girls from Kenilworth: 5
    - Girls from Wilmette: 10
    - Boys from Glencoe: 4
    - Boys from Kenilworth: 3
    - Girls from Glencoe: 8

• Probability of selecting a student from Kenilworth:
      - A) 0.132
      - B) 0.2
      - C) 0.211 (Correct)
      - D) 0.32
    - This probability encompasses the total number of students from Kenilworth versus all other students.

5. Mutual Exclusivity and Events

• Consider the outcomes of tossing a quarter four times:   - Total outcomes: 16
    - Outcomes include HHHH, HHHT, HHTH, HHTT, HTHH, HTHT, HTTH, HTTT, THHH, THHT, THTH, THTT, TTHH, TTHT, TTTH, TTTT.

• Define events:   - A: First two tosses are heads (HH).
    - B: First and last toss are the same (either HH or TT).   - Are events A and B mutually exclusive?     - A) Yes (Correct)
      - B) No
    - Exploring mutual exclusivity helps in understanding how events can coexist or exclude one another in probability.

Problem Solving and Calculations

6. Drinking Habits Probability

• A table of drinking habits by gender:
    - Men:
      - Non-drinker: 135
      - Regular Drinker: 47
      - Heavy Drinker: 5
      - Total: 187
    - Women:
      - Non-drinker: 187
      - Regular Drinker: 21
      - Heavy Drinker: 15
      - Total: 223
    - Total: 410

• Probability that a randomly selected student is a man or a non-drinker:
    - A) 0.947
    - B) 0.942 (Correct)
    - C) 0.834
    - D) 0.912
    - Solutions here involve combining the probabilities of selecting either category correctly.

Card Probability Questions

• Dealing from a Deck:

1. Probability of not dealing a spade:
       - A) $rac{3}{4}$ (Correct)
       - B) $rac{4}{13}$
       - C) $rac{2}{5}$
       - D) $rac{1}{4}$

Special Conditions and Events

8. Probability of drawing an ace or a 7:
     - A) $rac{4}{13}$ (Correct)
     - B) $rac{13}{2}$
     - C) 8
     - D) $rac{2}{13}$

9. Probability of being dealt a King first and a Queen second:
     - A) $rac{4}{663}$
     - B) $rac{1}{663}$ (Correct)
     - C) $rac{13}{102}$
     - D) $rac{2}{13}$

General Addition Rule in Probability

10. For randomly selected population:

• P(A) = 0.47, P(B) = 0.29, P(A & B) = 0.15

• Find P(A or B):
    - A) 0.91
    - B) 0.47
    - C) 0.61 (Correct)
    - D) 0.76

Multiplication in Probability

11. Probability of drawing two black cards without replacement:
      - A) 0.255
      - B) 0.000377
      - C) 0.245 (Correct)
      - D) 0.490

Conditional Probability

12. Find P(A | (not B)) for:

• A: diamond drawn; B: club drawn:
    - A) 0.25
    - B) 0
    - C) 0.333
    - D) $rac{1}{4}$ (Correct)

Contingency Tables

13. Rounds for voting in 1984:
      - Compute P(Democrat | Northeast):
        - A) 0.442 (Correct)
        - B) 0.241
        - C) 0.406
        - D) 0.098

14. Joint frequency of blood type and sex. Independence evaluation:
      - A) Yes
      - B) No (Correct)

Tree Diagram Responses

15. Two chips from a bag:
      - Contents: 5 red, 7 blue
      - Construct a tree diagram illustrating the probabilities associated with picking red and blue chips in succession, which aids in visualizing decision outcomes.

Probability Distribution

16. Probability distribution for tutor seeing students:

• Develop the probability distribution across x values (0, 1, 2, 3, 4), including methods for understanding how probabilities can sum to 1, with emphasis on the calculated distribution to ascertain missing probability values.

Expected Values

17. In a game with defined outcomes:
      - Analyze expected value calculated through defined win/loss outcomes, reinforcing decision-making expectations in statistical contexts.

Binomial and Poisson Distributions

19. Binomial Probability:
      - For n = 5, p = 0.3, calculate P(X = 3) by utilizing the binomial probability formula to analyze discrete distributions.

20. Probability for population under 25:
      - Summarize probabilities focusing on various age categories, ensuring comprehensive understanding of sampling techniques and their implications.

21. Using Poisson for specific events:
      - Analyze the mean number of calls received to determine the distributional characteristics and their application in real-world scenarios.

Test Key

• 1) B

• 2) B

• 3) D

• 4) C

• 5) B

• 6) D

• 7) A

• 8) D

• 9) A

• 10) C

• 11) C

• 12) C

• 13) A

• 14) A

• 15) B

• 16) C

• 17) A

• 18) B

• 19) C

• 20) A

• 21) C

• 22) D

• 23) C

• 24) C

• 25) D