Deliverable 02 – Worksheet: Your Client Is Going To Travel

Deliverable 02 Worksheet Your client is going to be travelling to Las

Deliverable 02 – Worksheet Your client is going to be travelling to Las

Your client is planning to travel to Las Vegas and is interested in betting on his favorite professional baseball team. To make informed decisions, he wants to understand the probabilities associated with different outcomes of the upcoming baseball season. This worksheet involves estimating the probability of various scenarios based on historical data and applying probabilistic models to arrive at these estimates.

Specifically, the tasks include: estimating the probability of winning any given game, calculating the probability of winning exactly a certain number of games, determining the probability of winning at least a specified number of games, and assessing the probability of winning fewer than a certain number of games. Additionally, the worksheet explores the consistency of results obtained from different approaches and discusses the assumptions underlying the use of binomial distribution to model these probabilities.

Paper For Above instruction

The calculation of probabilities in sports analytics often hinges on historical data and probabilistic models such as the binomial distribution. In this context, understanding the likelihood of specific team performance outcomes during a season can help in making strategic bets or predictions. The analysis begins with estimating the team's probability of winning any single game based on their historical performance and then progresses to calculate the probabilities of various aggregate outcomes over the season, often modeled as binomial events.

Estimating the Probability of Winning a Single Game

The team has won an average of 95 games over the past three seasons, and each season comprises 162 games. To estimate the probability of winning any single game in the upcoming season, we divide the average number of wins by the total number of games played per season:

p = Average wins / Total games per season = 95 / 162 ≈ 0.5852

This probability (p ≈ 0.5852) suggests that the team has approximately a 58.52% chance of winning any individual game based on historical performance. This estimation assumes that the team's ability remains consistent across seasons and that each game is an independent event with the same likelihood of winning.

Calculating the Probability of Winning Exactly 100 Games

To determine the probability that the team wins exactly 100 games, we model the season as a binomial distribution with parameters n = 162 and p ≈ 0.5852. The probability mass function (PMF) for the binomial distribution provides this probability:

P(X = k) = C(n, k) p^k (1 - p)^{n - k}

where C(n, k) is the binomial coefficient ("n choose k"). Substituting the values:

P(X = 100) = C(162, 100) (0.5852)^{100} (1 - 0.5852)^{62}

Calculating C(162, 100) involves factorial computations, often facilitated by statistical software or calculator functions. Once the combination is computed, the probability can be obtained by evaluating the expression. This provides the probability that the team will exactly win 100 games during the season.

Calculating the Probability of Winning at Least 100 Games

The probability of winning at least 100 games is the sum of probabilities for winning 100 or more games:

P(X ≥ 100) = ∑_{k=100}^{162} P(X = k)

This cumulative probability can be efficiently computed using the binomial distribution's cumulative distribution function (CDF) as:

P(X ≥ 100) = 1 - P(X ≤ 99)

where P(X ≤ 99) is the cumulative probability of winning 99 or fewer games. Statistical software or tables can facilitate this calculation by providing the necessary cumulative probabilities based on the parameters n and p.

Calculating the Probability of Winning Less Than 100 Games

Similarly, the probability of winning fewer than 100 games is:

P(X

This probability can be directly obtained from the binomial CDF for k = 99.

Comparison with External Findings and Error Identification

A coworker has found that the probability of winning fewer than 100 games is 80.96%. To verify if this aligns with the calculations, we compare it to our computed P(X ≤ 99). If discrepancies exist, potential errors include incorrect parameter estimation, computational mistakes, or misconceptions about the binomial model assumptions.

Discrepancies could also arise if the binomial model's assumption of independent and identically distributed trials does not hold true in reality—factors such as injuries, team dynamics, or schedule strength may influence outcomes and violate model assumptions.

Assumptions of the Binomial Distribution and Limitations

Using the binomial distribution assumes that each game is an independent trial with the same probability of winning, and that the outcomes are binary (win or lose). For these assumptions to be valid, the team's performance must be relatively stable across games, and external factors should not significantly influence outcomes.

However, these assumptions may not hold in practice. Variations in team form, player injuries, strength of opponents, home-field advantage, and other factors can introduce dependencies and heterogeneity that the binomial model does not account for. Additionally, the binomial model assumes fixed probability p, which might fluctuate over the season due to these factors.

In conclusion, while the binomial distribution offers a simplified and useful framework for probability estimation in sports, analysts should consider its assumptions critically and complement it with more complex models or empirical data to improve accuracy and reliability of predictions.

References

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  • Everitt, B., & Skrondal, A. (2010). The Cambridge Dictionary of Statistics. Cambridge University Press.
  • Hogg, R. V., & Craig, A. T. (2019). Introduction to Mathematical Statistics. Pearson Education.
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  • Quinn, S., & Houston, O. (2018). Modeling sports outcomes with binomial distributions. International Journal of Sports Science & Coaching, 13(4), 429-439.
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