## How to Assess Fairness in Football Using Probability

Probability is a statistical technique used to determine the likelihood of an event to happen, usually displayed as a number between 0 and 1. If an event has a probability of 0, then it is considered unlikely to happen while that with a possibility of 1 is considered likely to happen. If an event has a probability of 0.5, then it is considered to have equal odds of occurrence. Probability is applied in many disciplines today including weather prediction, astrophysics, medicine, music, and computer programming.

### Sampling

This memorandum details the procedures and results of a study conducted to determine test whether the quarter used by National Football League (NFL) is fair and if the quarter will remain fair after the weight is added to either side. The results of the unweighted quarter resulted in 28 heads and 22 tails; no significant bias was reported for this analysis. The results of the second analysis with a weighted quarter resulted in 18 heads and 32 tails; a significant bias was reported for this analysis.

The coin flip is one of the oldest decision-making tactics in history. It has been widely used in games and sports to determine who kicks off the game. A study by Levitt (2016) found that people who use a coin toss to decide on an important change are more likely to follow through with that decision, and are more satisfied with their decisions. The coin flip is an ideal decision-making strategy since the coin is equally likely to show heads or tails after one flip. This is usually the case unless the coin used for the flip is modified by shaving one side of the coin, adding weight to one side, or smearing something sticky on one side of the coin. In such cases, the coin becomes bias and can’t be trusted to make fair decisions.

This memorandum will detail the study procedures and results of the study to test the coin that the National Football League (NFL) will use in next year’s Super Bowl to determine who kicks off. The objective of the study is to determine whether the coin is fair. The study will also assess whether the coin would remain fair if an unscrupulous person or persons somehow managed to add some weight to one side of the coin or the other.

The layout of this memorandum is as follows: Section 2 discusses the study design, the specifics of how the data was collected and recorded, Section 3 will contain the description and summary of the data collected. Section 4 will give details regarding the statistical methods used for the analysis, the hypotheses, and the results of the analysis. Conclusions made from the study will be detailed in Section 5. Finally, Section 6 will contain the data used for the analysis. For more information on how to collect sample data, hire our sampling assignment help providers.

### Coin Flip Exercise

The data used for this analysis were collected from a coin flip exercise. We obtained a quarter and tossed the coins 50 times while recording the outcome at every toss. The outcomes were recorded as H for Heads and T for Tails. This would be determined by checking which side the quarter was showing when it lands, after the flip. If the quarter lands on Heads after a flip, the next flip is started when the Heads side is facing up. Similarly, if the quarter lands on ‘Tails’ after a flip, the next flip is started when the tail side is facing up. We performed the quarter flip 50 times and recorded the outcomes at every event. The data collected from this exercise is in Appendix 1.

To test whether the coin would remain fair if an unscrupulous person or persons somehow managed to add some weight to one side of the coin or the other, we performed a second toss exercise with some weight added to the head side. To add weight to the quarter, we obtained and smeared a small amount of modeling clay on the heads side of the quarter. This biased coin was tossed 50 times and the outcome recorded as done with the first toss exercise. The data collected from this exercise is in Appendix 2.

The data collection exercise resulted in 50 observations and two variables. The first variable contained the toss outcomes for a fair toss – the unmodified quarter. The table below summarizes the outcomes of the unbiased coin flip exercise.

Heads side came back facing up 28 times out of the 50 times the fair quarter was tossed. The tail side came back facing up only 22 times out of the 50 tosses performed.

The table below presents the outcome of the flip exercise with the biased coin with modeling clay smeared on the heads side. The heads side came back facing up only 18 times out of the 50 tosses performed.

### Hypothesis Testing

We tested two research questions using the data we obtained.

1. If we toss a fair quarter 50 times and obtained 28 heads and 22 tails, is the quarter fair?

2. If we toss a quarter which has been weighted on the heads side and obtained 18 heads and 32 tails, is the coin biased towards the tail?

To address these questions, we constructed this hypothesis:

Null hypothesis (Ho): The coin is fair i.e. p = 0.5

The alternate hypothesis (Ha): The coin is biased i.e. p ≠ 0.5

If the quarter is fair, we would see a 50% of the tosses ending with Heads facing up and 50% of the tosses ending with Tails facing up. If the quarter is biased, we would see one outcome being more frequent than the other.

We performed a one-sample proportions z test to determine whether there was a significant difference in the proportion of the Heads outcomes and the Tails outcomes.

We selected a level of significance of 0.05. The significance level is the measure of the strength of the evidence that must be present in our sample before rejecting the null. Our chosen value of 0.05 indicates a 5% risk of concluding that a difference exists between the two proportions when there is no actual difference.

The formula of the Z test used to acquire the test statistic is:

Where z = Test statistics we want to obtain,

n = Sample size

Po = Null hypothesized value

p^ = Observed proportion

We have a sample size of 50 and the null hypothesized value is 0.5.

For the fair toss, we will use the observed proportion of Heads i.e. 28/50 = 0.56.

The p-value associated with this analysis is 0.3961. Since the p-value is greater than 0.05, we fail to reject the null hypothesis. This means that there is no significant between the proportions of heads outcome (0.56) and the expected proportion of 0.5 if the quarter were unbiased.

For the weighted quarter, we will use the observed proportion of Tails i.e. 32/50 = 0.64.

The p-value associated with this analysis is 0.0477. Since the p-value is less than 0.05, we reject the null hypothesis. This means that there is a significant between the proportions of tails outcome (0.64) and the expected proportion of 0.5 if the quarter were unbiased. To get expert assistance with this topic, reach out to us for hypothesis testing assignment help.

### Conclusion

We performed a coin flip analysis with a quarter that hadn’t been modified. This resulted in 28 heads and 22 tails. The results of the analysis indicated that there was no significant difference between the proportion of Heads and the expected proportion of 50%. We conclude that the unmodified quarter is fair, and is, therefore, fit for use by the NFL. The analysis of the weighted quarter resulted in 18 heads and 32 tails. The results of the one-sample proportion z test indicated a significant difference between the proportion of tails and the expected proportion of 50%. When some weight was added on the heads side, the side became heavier and hence more likely to return facing down, resulting in more outcomes of tails facing up. Based on this result, we conclude that a quarter doesn’t remain fair after some weight is added to one side of the quarter.