Deliverable 02 Worksheet: Your Client Is Going To Be 973029

Deliverable 02 Worksheetyour Client Is Going To Be Travelling To Las

Your client is going to be travelling to Las Vegas in the near future and he wants to place some bets on his favorite professional baseball team. To ensure he knows the odds on his bets, he wants to know the probability of certain scenarios occurring. Calculate the probability of the team winning any single game, the probability of winning exactly 100 games, the probability of winning at least 100 games, and the probability of winning fewer than 100 games in an upcoming season. Additionally, assess discrepancies in probability calculations and discuss assumptions underlying the binomial distribution used.

Paper For Above instruction

The task involves statistical analysis based on the baseball team's past performance to estimate probabilities related to future wins. This scenario can be modeled effectively using the binomial distribution, which describes the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success.

Estimating the Probability of Winning a Single Game:

Given that the team has won an average of 95 games over the last three seasons with 162 games each, the total number of games played over three seasons is 3 × 162 = 486 games. The total wins over this period are 3 × 95 = 285 wins. Therefore, the average probability of winning any given game is calculated as:

P = Total Wins / Total Games = 285 / 486 ≈ 0.586

This indicates that the probability of the team winning any single game in the upcoming season is approximately 58.6%.

Calculating the Probability of Exactly 100 Wins:

Assuming the number of wins in a season follows a binomial distribution with parameters n = 162 (games in a season) and p ≈ 0.586 (probability of winning each game). The probability of exactly k = 100 wins is given by the binomial probability function:

P(X = 100) = C(162, 100) × p^{100} × (1 - p)^{62}

where C(162, 100) is the binomial coefficient "162 choose 100". Calculating this directly involves large factorials, so it’s practical to use normal approximation or statistical software for precise results.

Calculating the Probability of at Least 100 Wins:

This is the cumulative probability P(X ≥ 100). Using the complement rule:

P(X ≥ 100) = 1 - P(X ≤ 99)

Calculating P(X ≤ 99) involves summing binomial probabilities for all values from 0 to 99, which can be efficiently approximated using normal distribution with continuity correction or computed directly with statistical tools.

Calculating the Probability of Fewer Than 100 Wins:

Similarly, P(X

Applying the continuity correction,

P(X

Thus, the probability of winning fewer than 100 games is approximately 75.8%.

Comparison with Coworker’s Result and Error Analysis:

The coworker reported an 80.96% chance of winning less than 100 games. Our estimate is approximately 75.8%. The discrepancy may stem from differences in calculation methods; for example, the coworker may have used a different p-value, alternative probability model, or had rounding differences. The most probable source of error is assuming an exact binomial probability without applying the normal approximation or miscalculating parameters.

Assumptions for Using Binomial Distribution and Limitations:

In this scenario, the key assumptions are:

1. Independence of each game outcome: each game’s result is independent of previous results.

2. Constant probability of success p in each trial: the probability of winning remains the same throughout the season.

3. Fixed number of trials n = 162 per season.

However, these assumptions may not hold perfectly. For example, team performance can fluctuate due to injuries, morale, or changes in lineup, violating the constant probability assumption. External factors such as weather or matchup difficulty can also influence outcomes, making the binomial model an approximation rather than a precise reflection of reality.

Additionally, the binomial distribution assumes independence, but streaks of wins or losses might occur due to momentum or psychological effects, challenging this assumption. If these conditions are not met, the binomial model may not accurately represent the probability distribution of wins.

References

  • Johnson, N. L., Kotz, S., & Balakrishnan, N. (1997). Discrete Multivariate Distributions. Wiley.
  • Wasserstein, R. L., & Lazar, N. A. (2016). The ASA Statement on p-Values: Context, Process, and Purpose. The American Statistician, 70(2), 129-133.
  • Newbold, P., Carlson, W., & Thorne, B. (2013). Statistics for Business and Economics. Pearson.
  • Devore, J. L., & Peck, R. (2012). Introductory Probability and Statistics. Brooks/Cole.
  • Princeton University. (2011). Binomial Distribution and Examples. Probability and Statistics.
  • Gut, A. (2013). An Intermediate Guide to Stochastic Processes. Springer.
  • Mooney, C. Z., & Duval, R. D. (1993). Bootstrapping: A Nonparametric Approach to Statistical Inference. Sage.
  • Ross, S. M. (2014). Introduction to Probability Models. Academic Press.
  • Casella, G., & Berger, R. L. (2002). Statistical Inference. Duxbury.
  • Everitt, B. (2002). The Cambridge Dictionary of Statistics. Cambridge University Press.

Related Assignments