5.1, 5.2

NFL Overtime Coin Toss Statistics

• Overtime Rules (Pre-2012): After a tie at the end of regulation, a coin toss determined which team got the choice to receive the ball or to kick off.

• Coin Toss Outcomes:

  • 98.1% of teams that won the coin toss chose to receive the ball.

Historical Data on Overtime Games

• Games Analyzed: Between 1974 and February 2011, there were 477 overtime games.

• Final Outcomes:

  • 17 ended in a tie, leaving 460 games with a clear outcome.

• Wins Distribution:

  • 252 games won by the team that won the coin toss (54.8%)

  • 208 games won by the team that lost the coin toss.

• Importance of Scoring: During this period, the first team to score won the game regardless of the type of scoring.

Statistical Significance Question

• Analytical Focus: Is the 252 wins by teams that won the coin toss statistically significant?

• Statistical Concepts: Understanding statistical significance is crucial as this ties into probability distributions discussed in prior chapters.

Probability Distributions

• Definition: A probability distribution provides a function that specifies the probability of obtaining the possible values that a random variable can take.

• Example of Two Coin Tosses:

  • Prior analysis of two coin tosses introduced frequency distributions and theoretical probabilities.

• Constructing Probability Distribution:

  • For two coin flips, the outcomes can be: 0 heads, 1 head, or 2 heads, with respective probabilities.

Random Variable and Probability Distribution Definitions

• Random Variable: Denoted by X, it represents the outcomes resulting from chance processes.

• Probability Distribution: Provides probabilities for each possible value of the random variable.

Discrete vs Continuous Random Variables

• Discrete Random Variable: Has countable outcomes (e.g., number of heads in coin tosses).

• Continuous Random Variable: Has infinitely many possible values (e.g., measurements).

Requirements for a Probability Distribution

1. A numerical random variable (not categorical).

2. Sum of all probabilities must equal 1.

3. Each individual probability should be between 0 and 1, inclusive.

Example of a Probability Distribution

• Random Variable: Number of heads from flipping two coins.

• Probability Values:

  • 0 heads: 0.25

  • 1 head: 0.50

  • 2 heads: 0.25

Calculating Mean and Standard Deviation

• Mean: The expected value calculated from the probability distribution.

• Variance: Measurement of the distribution's spread, from which you can find standard deviation (by taking the square root).

Statistical Significance Determination

• Range Rule of Thumb: Knowing whether a value is significantly high or low based on its deviation from the mean.

• Significant High: Value is significantly high if it is more than 2 standard deviations above the mean.

• Analysis of NFL Wins: If the win count of 252 is significantly above expected values, it hints at the advantage of winning a coin toss.

Application in Real Scenarios

• Real-World Example: Statistics surrounding coin tosses can be applied in various fields, illustrating how numerical analysis influences decision-making.

• Expected Value: Often a focus in finance and insurance decisions.

Two Categories in Probability Distributions

• Binomial Probability Distribution: Involves scenarios with two outcomes (success and failure).

• Example Calculation: Probability of adults knowing Twitter across fixed trials with independent chances.

Summary of Statistical Concepts

• Expected Value in Probability: Used to quantify outcomes in scenarios studied, including advantages historically noted in NFL statistics.

• Visual Understanding: Created and analyzed distribution figures to comprehend significance and value probabilities.