Psy 520 Graduate Statistics Topic 3 Probability Project Dire

Psy 520 Graduate Statistics topic 3 Probability Project directions

Use the following information to complete the assignment. While APA format is not required for the body of this assignment, solid academic writing is expected, and documentation of sources should be presented using APA formatting guidelines, which can be found in the APA Style Guide, located in the Student Success Center.

Several misconceptions about probability are discussed, including assumptions that all events are equally likely, later events may influence earlier ones, sample size is irrelevant when calculating probability, and others. All of these statements are incorrect, and the task is to explain why, utilizing probability rules from the course.

Specific statements to analyze include:

1. I have flipped an unbiased coin three times and got heads; it is more likely to get tails the next time I flip it.
2. The Rovers play Mustangs, and since the game can result in a win, loss, or draw, the probability of the Rovers winning is 1/3.
3. I roll two dice and sum the results; the probability of obtaining a total of 6 is 1/12 because there are 12 different outcomes, and 6 is one of them.
4. Mr. Purple needs a major operation with a 90% chance of full recovery if he undergoes the procedure.
5. I flip two coins; the probability of getting one heads and one tails is 1/3 because the possible outcomes are heads-heads, heads-tails, or tails-tails.
6. The number 13 is considered unlucky, so you are less likely to win raffles with ticket number 13 than with other numbers.

Additionally, there is a detailed narrative about Chipotle Mexican Grill’s crisis in 2015 involving foodborne illness outbreaks, health investigations, and their impact on sales and stock prices, which appears unrelated to specific probability questions but may serve as contextual background or examples for probability applications.

Paper For Above instruction

The misconceptions about probability present common errors in understanding how random events and probabilities function according to fundamental rules. These misconceptions lead to inaccurate assumptions that can affect decision-making, interpretation of data, and understanding of randomness. Correcting these errors requires a clear understanding of the basic probability principles, including the notion that each event's outcome is independent of previous occurrences in the case of independent events, and that the probability of combined events depends on the outcomes' combinations, not just the number of possible outcomes.

Analysis of Individual Statements

1. Coin Flips and the Gambler’s Fallacy

The assertion that after flipping a coin three times and obtaining heads, the next flip is more likely to be tails is a classic example of the gambler’s fallacy. Each flip of an unbiased coin is an independent event, meaning the outcome of previous flips does not influence the next. Probability rules specify that the chance of heads or tails remains 0.5 regardless of previous results. Although it might seem intuitive that tails are "due" after multiple heads, this is a misconception. The probability of heads or tails on any flip is always 0.5, independent of prior outcomes (Kahneman & Tversky, 1972).

2. Probability of a Win in a Three-Outcome Game

The statement that the probability of the Rovers winning is 1/3 because they can win, lose, or draw ignores the real-world probabilities influenced by team strength, historical data, and other factors. For a purely theoretical model assuming equally likely outcomes, the probability of any specific outcome (e.g., a win) could be 1/3. However, actual probabilities are rarely equal and should be based on empirical data rather than assumptions (Brams & Taylor, 1996). Therefore, the assumption of equal likelihood is an oversimplification and likely incorrect in practical terms.

3. Dice Roll and Summation Probabilities

Calculating the probability of rolling a total of 6 with two dice involves understanding the total possible outcomes. Each die has six faces, leading to 36 total combinations. The outcomes summing to 6 are (1,5), (2,4), (3,3), (4,2), and (5,1). There are 5 favorable outcomes, so the probability is 5/36, approximately 0.139. The statement that the probability is 1/12 (about 0.083) because there are 12 outcomes and 6 is 'one of them' is mistaken because it conflates the total number of outcomes with the number of favorable outcomes. Correct calculation considers the specific combinations (Kendall & Stuart, 1973).

4. Probabilistic Recovery Rate

When it is stated that Mr. Purple has a 90% chance of recovery, this reflects a probability based on medical data and clinical studies. It presumes that outcomes are probabilistically independent and the patient’s likelihood of recovery is consistent with the statistical rate. This probability is not affected by previous outcomes but is an estimate based on population data. Hence, the statement about his chance of recovery aligns with probability principles assuming the data is accurate and relevant (Hassan et al., 2021).

5. Probability of Different Coin Outcomes

The claim that flipping two coins yields a 1/3 chance of heads and tails due to outcomes including heads-heads, heads-tails, or tails-tails demonstrates a misconception. The actual discrete outcomes are: HH, HT, TH, TT, totaling four equally likely outcomes when flipping two coins, assuming fairness and independence. The probability of getting one head and one tail, which occurs with HT or TH, is 2/4 or 1/2. The initial assertion is incorrect because it miscounts outcomes and misapplies probability calculation (Feller, 1968).

6. The "Unlucky" Number Effect on Raffle Outcomes

The belief that ticket number 13 is less likely to win because it is considered unlucky is a superstitious misconception. Probability is not affected by beliefs, superstitions, or numbers' cultural significance. The chance of winning with ticket 13, assuming a fair draw, is identical to any other ticket. This aligns with the principle that each ticket has an equal probability of winning, regardless of superstition or superstition-related biases (Elliott, 1990).

Implications of Misconceptions in Real-World Contexts

A real-world example highlighting the importance of understanding probability correctly is the case of Chipotle’s outbreak investigations. Misjudging probabilities regarding food safety contamination, or assuming equal likelihood of contamination across ingredients without evidence, can lead to flawed decision-making. In food safety management, applying accurate probability models allows companies to better allocate resources and mitigate risks (Huang et al., 2017).

Similarly, consumers' misconceptions about probability, such as believing in "lucky" numbers or that certain outcomes are "due," can influence behaviors and perceptions in ways that are unsupported by the actual probability theory. Recognizing independence and the nature of random events helps avoid cognitive biases that lead to bad decisions and ineffective strategies.

Conclusion

Understanding and accurately applying probability rules dispels misconceptions that can mislead individuals in both everyday and professional settings. Independence of events, correct calculation of total outcomes, and the understanding that superstitions do not influence randomness are foundational principles. As illustrated through the analysis of specific statements, these principles are essential for interpreting real-world phenomena accurately, whether in gambling, sports, or food safety management.

References

• Brams, S. J., & Taylor, A. D. (1996). Fair division: From cake-cutting to dispute resolution. Cambridge University Press.
• Elliott, A. (1990). Superstitions: A social history. Holt, Rinehart and Winston.
• Feller, W. (1968). An introduction to probability theory and its applications (Vol. 1). Wiley.
• Hassan, M., Khalil, M., & Abdelrahman, M. (2021). Medical prognosis and the application of probability in healthcare. Journal of Medical Practices & Research, 9(2), 45-53.
• Huang, H., Meng, Q., & Li, R. (2017). Risk assessment in food safety management: A probabilistic approach. Food Control, 78, 328–336.
• Kahneman, D., & Tversky, A. (1972). Subjective probability: A judgment of representativeness. Cognitive Psychology, 3(3), 430–454.
• Kendall, M., & Stuart, A. (1973). The advanced theory of statistics (Vol. 2). Charles Griffin & Co.