Assignment 2 Outcomes And Outcome Spaces For The Expe 853797

Assignment 2 Outcomes And Outcome Spacesfor The Experiments Defined I

For the experiments described, determine the set of all possible outcomes (the outcome space). Indicate whether the outcomes are equally likely and if they are mutually exclusive. The specific experiments include:

  1. Tossing a coin three times and recording heads or tails on each toss.
  2. Recording the number of accidents in NYC on the day you complete the homework, assuming accidents occur randomly.
  3. Recording the height of a randomly selected adult male to the nearest inch.
  4. Injecting a drug into 5 randomly selected diseased mice and recording how many are cured after one week.
  5. Calculating the number of outcomes in an experiment involving one, two, and N coin tosses, respectively.
  6. Listing outcomes for specific events based on heights of adult males.
  7. Listing outcomes for specific events based on the number of mice cured in the experiment with 5 mice.
  8. Listing outcomes for specific events based on the results of three coin tosses.
  9. Calculating the probabilities of the events in Problem 8.

Paper For Above instruction

The analysis of outcome spaces and their properties is fundamental in understanding probability theory. In this paper, we explore various experiments, determining their outcome sets, whether outcomes are equally likely, and whether they are mutually exclusive. Each experiment offers insight into different probabilistic structures, from simple coin tosses to more complex biological and social phenomena.

1. Toss a coin three times and record heads or tails on each toss

The outcome space for three coin tosses comprises all possible sequences of heads (H) and tails (T). There are 2 options for each toss, resulting in 2^3 = 8 total outcomes: {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}. These outcomes are mutually exclusive because each sequence is distinct, and they are equally likely assuming a fair coin, each outcome with probability 1/8.

2. Record the number of accidents in NYC on the day you complete this homework

The outcome space here is the set of non-negative integers, {0, 1, 2, 3, ...}, representing the count of accidents. Typically, these counts are modeled as a Poisson distribution, which suggests outcomes are not equally likely; smaller counts tend to be more probable than larger counts. The outcomes are not mutually exclusive, as each outcome is a distinct number of accidents.

3. Record the height of a randomly selected adult male to the nearest inch

The outcome space is a set of possible heights, which can be considered discrete, such as {60 inches, 61 inches, ..., 80 inches} depending on the population. Outcomes are generally not equally likely; for example, certain heights have higher frequencies based on population distribution. These outcomes are mutually exclusive, as a person cannot have two different heights simultaneously.

4. Inject a drug into 5 diseased mice and record how many are cured after one week

The outcome space consists of the integers from 0 to 5, representing the number of cured mice in each trial: {0, 1, 2, 3, 4, 5}. Outcomes are mutually exclusive and, depending on the experimental context, may not be equally likely; probabilities can vary based on the drug's efficacy.

5. Outcomes in coin toss experiments with one, two, and N tosses

For one toss, outcomes are {H, T} (2 outcomes). For two tosses, outcomes are {HH, H T, T H, T T} (4 outcomes). For N tosses, outcomes are all sequences of length N of H and T, totaling 2^N outcomes. All outcomes are mutually exclusive, and assuming a fair coin, they are equally likely with probability 1/2^N each.

6. Outcomes based on height for event A and B

Event A: Selectees less than 6’ tall (less than 72 inches) - outcomes: heights in the set {60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71}.

Event B: Selectees between 5’ and 6’5”, inclusive (60 to 77 inches).

The combined outcomes for these events are subsets of the overall height outcome space, with overlaps between A and B where heights are between 60 and 71 inches.

7. Outcomes for the curing experiment for mice

Event A: All mice are cured: {5} mice cured.

Event B: More than half the mice are cured: {3, 4, 5}.

Event C: An even number of mice cured: {0, 2, 4}. These event outcomes are subsets of the total outcomes {0, 1, 2, 3, 4, 5} and are mutually exclusive in their definitions.

8. Outcomes for coin toss events

Event A: All tosses show the same face: {HHH, TTT}.

Event B: At least one heads: Outcomes include all sequences except TTT.

Event C: At most one heads: outcomes {HTT, THT, TTH, TTT}. These events are not all mutually exclusive (for example, the event "all same face" overlaps with "at least one heads" in the case of HHH). Their probabilities depend on the fairness of the coin.

9. Probabilities of events in Problem 8

For a fair coin, probability of Event A (all same face): 2 outcomes (HHH, TTT) out of 8 total outcomes, so P = 2/8 = 1/4.

Probability of Event B (at least one heads): 1 minus the probability of no heads (TTT), so P = 1 - 1/8 = 7/8.

Probability of Event C (at most one heads): outcomes {HTT, THT, TTH, TTT}, 4 outcomes, so P = 4/8 = 1/2.

Conclusion

Exploring the various outcome spaces reveals the diversity of probabilistic models, from simple equally likely scenarios like coin tosses to more complex, uneven distributions like accident counts or biological responses. Recognizing mutual exclusivity and likelihood assumptions is crucial in designing experiments and interpreting results. Such foundational understanding enhances the capacity to analyze uncertainties across multiple disciplines, including statistics, biology, and social sciences.

References

  • Casella, G., & Berger, R. L. (2002). Statistical Inference (2nd ed.). Duxbury Press.
  • Devore, J. L. (2015). Probability and Statistics for Engineering and the Sciences (8th ed.). Cengage Learning.
  • Ross, S. M. (2014). Introduction to Probability and Statistics (11th ed.). Academic Press.
  • Wasserman, L. (2004). All of Statistics: A Concise Course in Statistical Inference. Springer.
  • Jaynes, E. T. (2003). Probability Theory: The Logic of Science. Cambridge University Press.
  • Hosmer, D. W., Lemeshow, S., & Sturdivant, R. X. (2013). Applied Logistic Regression. Wiley.
  • Kennedy, P. (2008). A Guide to Econometrics (6th ed.). Wiley.
  • Lehmann, E. L., & Casella, G. (1998). Theory of Point Estimation. Springer.
  • Moore, D. S., McCabe, G. P., & Craig, B. A. (2012). Introduction to the Practice of Statistics (8th ed.). W. H. Freeman.
  • Van der Meer, A., & Frankel, S. (2017). Understanding Probability. Academic Press.

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