Imagine Yourself At A Fair Playing One Of The Midway Games

Imagine Yourself At A Fair Playing One Of The Midway Games Pick A

Imagine yourself at a fair playing one of the midway games. Pick a game and calculate the expected value and post your results along with how you calculated them. For example, you may decide to throw a basketball to try to win a $10 bear. You paid $2.00 for three shots. What is the expected value? (Please do not use this example in your answer) 2. In one sentence, summarize what you understand of the Central Limit Theorem. 4. se the mean and the standard deviation obtained from the last module and test the claim that the mean age of all books in the library is greater that 2005. Share your results with the class.

Sample Paper For Above instruction

Choosing a game at a fair and analyzing it from a probabilistic perspective can provide an interesting insight into expected values and the risks involved. For this exercise, I selected a ring toss game where the goal is to land a ring around a bottle neck to win a prize. In this game, the player pays $3 for three tosses, and the objective is to land as many rings as possible to increase the chances of winning a prize, such as a plush toy.

To compute the expected value of participating in this game, I first identified the probabilities and payoffs associated with the possible outcomes. Suppose the probability of successfully landing a ring around the bottle in a single toss is 0.2, based on observations at the game booth. The prize for winning is a plush toy valued at $15. Conversely, if the player does not win, they obtain no prize, but the cost paid for playing is $3 per attempt.

The expected value (EV) can be calculated by summing the products of each outcome’s payoff and its probability:

  • Winning the plush toy: The net gain is $15 minus the $3 cost, which equals $12. The probability of winning in one attempt is 0.2.
  • Not winning: The net gain is -$3, with a probability of 0.8.

Therefore, the expected value per game is:

EV = (Probability of win × Net gain if win) + (Probability of loss × Net loss if lose)

EV = (0.2 × $12) + (0.8 × -$3)

EV = $2.40 - $2.40 = $0

This calculation shows that the expected value for participating in the ring toss game is zero, indicating that, on average, the game is fair, and the player breaks even in the long run. This analysis highlights that despite the possibility of winning a prize, the game is designed such that the expected return for the player is neutral, considering the probabilities and prizes involved.

Understanding the Central Limit Theorem

The Central Limit Theorem states that, for a sufficiently large sample size, the sampling distribution of the sample mean will be approximately normally distributed, regardless of the shape of the population distribution.

Testing the Mean Age of Books in the Library

Using the mean and standard deviation obtained from the last statistical module, I tested the claim that the average age of all books in the library is greater than 2005. I formulated the hypotheses as follows:

Null hypothesis (H0): μ ≤ 2005

Alternative hypothesis (H1): μ > 2005

Assuming the sample mean age was 2010 with a standard deviation of 10 and a sample size of 30 books, I conducted a one-sample t-test. The test statistic is calculated as:

t = (Sample Mean - Hypothesized Mean) / (Standard Deviation / √n)

t = (2010 - 2005) / (10 / √30) ≈ 3.44

Referring to a t-distribution table with 29 degrees of freedom, a t-value of 3.44 corresponds to a p-value less than 0.01, indicating strong evidence to reject the null hypothesis. Therefore, we conclude that the mean age of the books in the library is statistically significantly greater than 2005, supporting the claim.

References

  • Casella, G., & Berger, R. L. (2002). Statistical Inference (2nd ed.). Duxbury.
  • Freund, J. E., & Williams, F. (2003). Modern Elementary Statistics (12th ed.). Pearson Education.
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