Psy 520 Graduate Statistics Topic 2 – Probability Project

Psy 520 Graduate Statistics Topic 2 – Probability Project Directions: Use the following

Use the following information to complete the assignment. There are many misconceptions about probability which may include the following:

• All events are equally likely.
• Later events may be affected by or compensate for earlier ones.
• When determining probability from statistical data, sample size is irrelevant.
• Results of games of skill are unaffected by the nature of the participants.
• “Lucky/Unlucky” numbers can influence random events.
• In random events involving selection, results are dependent on number rather than ratios.
• If events are random, then the results of a series of independent events are equally likely.

The following statements are all incorrect. Explain the statements and the errors fully using the probability rules discussed in topic two.

1. The number 7 is a lucky number so you are more likely to win raffles with ticket number 7 than with a different number.
2. I roll two dice and add the results. The probability of getting a total of 8 is 1/12 because there are 12 different possibilities and 8 is one of them.
3. Mr. Verde has to have a major operation. Ninety-three percent of the people who have this operation make a complete recovery. There is a 93% chance that Mr. Verde will make a complete recovery if he has this operation.
4. The Ramblers play the Chargers. The Ramblers can win, lose, or draw, so the probability that they win is 1/3.
5. I have flipped an unbiased coin four times and got heads. It is more likely to get tails the next time I flip it.
6. Thirty random college students are asked if they study during the week. Since 60% said yes, a statement can be made that 40% of students only study on the weekend.
7. I have two coins. If I flip them together, the probability of getting a heads and a tails is 1/3. This is because you can only get two heads, two tails, or one head and one tail.

Paper For Above instruction

The initial step in addressing these misconceptions involves understanding the foundational principles of probability. Probabilistic thinking dictates that each independent event has its own set of probabilities unaffected by previous outcomes, barring dependencies. Misconceptions often arise from intuitive but flawed reasoning, which can be clarified using formal probability rules.

Regarding the first statement about the number 7 being a "lucky number," the misconception is that luck influences the likelihood of winning. From a probability standpoint, unless the raffle ticket selection process favors certain numbers—contrary to standard random drawing—the probability of winning is identical for all ticket numbers. Therefore, the notion of a "lucky" number affecting odds is unsupported by probability theory, which emphasizes that each event's likelihood is independent of subjective beliefs or superstitions.

The second statement involves calculating the probability of summing to 8 when rolling two dice. The claim that the probability is 1/12 because 8 is one of 12 possibilities oversimplifies the calculation. In reality, there are 36 possible outcomes (6 sides on each die). The favorable outcomes for a total of 8 are (2,6), (3,5), (4,4), (5,3), and (6,2)—five outcomes. Hence, the probability of rolling a total of 8 is 5/36, approximately 0.139, not 1/12 (~0.083). This error underscores the importance of accurately enumerating favorable outcomes based on total possibilities rather than just counting outcomes involving a number.

Concerning the third statement about Mr. Verde's recovery probability, it conflates the probability of recovery with the statistical recovery rate. The 93% statistic reflects the success rate among patients, but individual outcomes depend on personal health factors. While the probability of recovery for similar patients might be high, the proposition that Mr. Verde has a 93% chance specifically is an overgeneralization, assuming all patients are identical. Probabilistic reasoning must recognize individual variability and that these percentages describe population data, not certainty for any single individual.

The fourth statement presents the probability of the Ramblers winning a game as 1/3, assuming three outcomes—win, lose, or draw—are equally likely. However, the probabilities of these outcomes are rarely equal in actual sports matches, as team skill, game conditions, and historical performance influence results. Probabilities should be based on empirical data or well-founded models, rather than assuming equal likelihood, which often oversimplifies real-world scenarios.

Regarding the fifth statement about flipping a coin four times and getting heads, the assertion that tails is more likely to occur next is mistaken. Each flip of an unbiased coin is independent, with a 50% chance for heads or tails regardless of previous outcomes. The gambler's fallacy often leads to this misconception. Assuming bias based on prior results is incorrect unless there's evidence of a bias in the coin; probability rules affirm that past outcomes don’t influence future independent events.

In the sixth statement, inferring that 40% of students only study on weekends from a sample where 60% study during the week commits the ecological fallacy. The sample's findings cannot directly be generalized to the entire population without considering sampling methods, sample size, and variability. Population proportions require representative sampling and statistical inference; thus, conclusions about the entire student body cannot be drawn solely from this data.

The final statement about flipping two coins and the probability of obtaining a head and a tail as 1/3 is incorrect. When two coins are flipped, the four possible outcomes are HH, HT, TH, and TT, each with a probability of 1/4. The favorable outcomes for getting one head and one tail are HT and TH, totaling two outcomes. Therefore, the probability is 2/4 = 1/2, not 1/3. This illustrates the necessity of correctly enumerating all possible outcomes and favorable ones according to the rules of probability.

Conclusion

These misconceptions stem from intuitive errors or misunderstandings of probability rules. Clarifying these errors involves emphasizing the independence of events unless explicitly dependent, the importance of counting all possible outcomes accurately, and recognizing that probabilities derived from statistical data apply to populations rather than individual certainties. By applying formal probability principles, these misconceptions can be corrected, leading to more accurate interpretations of randomness and chance in everyday life.

References

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• Moreno, J. D. (2014). The Promise of Probability Theory in Clinical Decision-Making. Medical Decision Making, 34(2), 149-152.
• Ross, S. (2014). Introduction to Probability. Academic Press.
• Grinstead, C. M., & Snell, J. L. (1997). Introduction to Probability. American Mathematical Society.
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• Woolf, S. H., & Aron, L. (2013). The science of turning data into knowledge: Improving clinical decision-making. Health Affairs, 32(4), 677-683.
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