Answer All Questions, Show Work, Ask Instructor For Clarific

Answer All Questions Show All Work Ask Instructor For Any Clarificat

Answer all questions. Show all work. Ask instructor for any clarification.

Paper For Above instruction

1. What is probability? (2 pts)

Probability is a measure of the likelihood that a specific event will occur. It is expressed as a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty. Mathematically, the probability of an event A is given by the ratio of favorable outcomes to the total number of possible outcomes, provided all outcomes are equally likely: P(A) = Number of favorable outcomes / Total number of outcomes.

2. The Porter family's probability of winning first prize at a raffle is 3/117. What is the probability that they will lose? Explain your answer. (4 pts)

The probability that the Porter family will lose the raffle is the complement of their winning probability. Since the probability of winning is 3/117, the probability of losing is 1 minus that value.

Calculation: P(loss) = 1 - P(win) = 1 - (3/117) = (117/117) - (3/117) = (114/117) = 38/39.

This means that the chances they will lose are quite high, with a probability of 38/39, which reflects that in most cases, they are unlikely to win.

Assignment 3 Questions 3 & 4

Use the image from the link to answer questions 3 and 4:

3a. What is the sample space for the color game wheel? (3 pts)

The sample space for the color game wheel comprises all possible outcomes when the wheel is spun. If the wheel has distinct color segments, then the sample space is the set of all colors present, such as {Red, Yellow, Blue, Green, etc.}. Each outcome represents the wheel landing on one color segment.

3b. What is the sample space of the cubic die? (3 pts)

The sample space for a standard six-sided die is the set of all possible outcomes of a single roll: {1, 2, 3, 4, 5, 6}.

3c. Are the sample spaces for each object fair or unfair? Explain (4 pts)

The fairness of a sample space depends on whether each outcome has an equal probability. For the die, assuming it is a standard fair die, each number from 1 to 6 has an equal probability of 1/6, making it a fair sample space. For the game wheel, if each segment is of equal size and the wheel is balanced, then each outcome (color) has an equal chance, and the sample space is fair. If the segments are unequal or the wheel is weighted, then the sample space is unfair.

Questions 4a–4d

4a. What is the probability of getting yellow on the game wheel? Explain (3 pts)

The probability of landing on yellow is the ratio of yellow segments to total segments on the wheel. If the wheel has, for example, 8 segments with 2 yellow segments, then P(yellow) = 2/8 = 1/4.

4b. What is the probability of getting blue on the wheel? Explain (3 pts)

Similar to yellow, if there are, for example, 3 blue segments out of 8, P(blue) = 3/8.

4c. What is the probability of not getting blue nor yellow? Explain (4 pts)

The probability of not getting blue or yellow is the complement of the probability of landing on either blue or yellow segments. Using the example, P(not blue or yellow) = 1 - P(blue or yellow) = 1 - (P(blue) + P(yellow)), assuming the events are mutually exclusive. If P(blue) = 3/8 and P(yellow) = 2/8, then P(not blue or yellow) = 1 - (3/8 + 2/8) = 1 - 5/8 = 3/8.

4d. If the game wheel is spun 20 times, how many times should we expect it to land on the blue space? Explain (5 pts)

The expected number of landings on blue is calculated by multiplying the probability of landing on blue by the number of spins. If P(blue) = 3/8, then expected occurrences = 20 × 3/8 = 20 × 0.375 = 7.5. Therefore, in 20 spins, we expect approximately 7 or 8 landings on blue, based on probability.

Questions 5b and 5c

5b. Explain Brian's chances of getting all questions right. (3 pts)

Each question has 4 choices, only one of which is correct. Since Brian guesses randomly, the probability of correctly answering one question is 1/4. The probability of getting all 5 questions correct is (1/4)^5 = 1/1024, indicating extremely low chances.

5c. He needs to get at least 3 correct to pass the test. Carefully explain Brian's chances of passing. (6 pts)

To find the probability that Brian passes, we consider the sum of probabilities of him getting exactly 3, 4, or 5 questions correct. Using the binomial probability formula:

P(k correct) = C(n, k) × (p)^k × (1 - p)^{n - k}

where n = 5, p = 1/4, and C(n, k) is the binomial coefficient.

Calculations:

  • Probability of exactly 3 correct: C(5, 3) × (1/4)^3 × (3/4)^2 = 10 × (1/64) × (9/16) = 10 × (9/1024) = 90/1024 ≈ 0.0879
  • Probability of exactly 4 correct: C(5, 4) × (1/4)^4 × (3/4)^1 = 5 × (1/256) × (3/4) = 5 × (3/1024) = 15/1024 ≈ 0.0146
  • Probability of exactly 5 correct: C(5, 5) × (1/4)^5 × (3/4)^0 = 1 × (1/1024) = 1/1024 ≈ 0.00098

Adding these probabilities: (90 + 15 + 1)/1024 = 106/1024 ≈ 0.1035, or about 10.35%. Therefore, Brian has approximately a 10.35% chance of passing the test.

Note on Question 6

Question 6 refers to an image from a link that was not provided. Without the image or additional details, it is not possible to answer this question accurately. If further information is supplied, the response can be appropriately addressed.

References

  • Bartholomew, D. J., Booth, K. N., Moustaki, I., & Galbraith, J. (2011). Are You Probable?: The Handbook of Probability and Evidence. Cambridge University Press.
  • Feller, W. (1968). An Introduction to Probability Theory and Its Applications, Volume 1. John Wiley & Sons.
  • Grinstead, C. M., & Snell, J. L. (1997). Introduction to Probability. American Mathematical Society.
  • Ross, S. M. (2014). Introduction to Probability Models. Academic Press.
  • Devore, J. L. (2015). Probability and Statistics for Engineering and the Sciences. Cengage Learning.
  • Krause, B. (2010). Probability For Dummies. John Wiley & Sons.
  • Moore, D. S., Notz, W. I., & Fligner, M. A. (2013). Statistics: Concepts and Controversies. W. H. Freeman.
  • Walpole, R. E., Myers, R. H., Myers, S. L., & Ye, K. (2012). Probability & Statistics for Engineering and the Sciences. Pearson.
  • Blitzstein, J., & Hwang, J. (2014). Introduction to Probability. Chapman & Hall/CRC.
  • Hogg, R. V., & Tanis, E. A. (2010). Probability and Statistical Inference. Pearson Education.

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