Coin Toss Fairness

Solution & Explanation

To determine if the coin can be considered fair, we need to analyze the probability of observing such an outcome (8 tails and 2 heads) under the assumption that the coin is fair. A fair coin implies that the probability of getting heads (p) is 0.5 and the probability of getting tails (1-p) is also 0.5.

Step 1: Understanding the Binomial Distribution

The binomial distribution models the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. In this context:

• n = 10 (total number of coin flips)
• x = 2 (number of heads, considered as "successes")
• p = 0.5 (probability of getting heads)

The probability of getting exactly x successes in n trials is given by the formula:

$P(X = x) = \binom{n}{x} p^x (1-p)^{n-x}$

Where:

• $\binom{n}{x}$ is the binomial coefficient, representing the number of ways to choose x successes from n trials, calculated as: $\binom{n}{x} = \frac{n!}{x!(n-x)!}$

Step 2: Calculate the Probability

For the given scenario (8 tails and 2 heads):

• x = 2
• n = 10
• p = 0.5

Plug these values into the formula:

$P(X = 2) = \binom{10}{2} (0.5)^2 (1-0.5)^{10-2}$

Calculate the binomial coefficient:

$\binom{10}{2} = \frac{10 \times 9}{2 \times 1} = 45$

Now calculate the probability:

$P(X = 2) = 45 \times (0.5)^2 \times (0.5)^8 = 45 \times (0.5)^{10} = 45 \times 0.0009765625 \approx 0.0439$

Step 3: Statistical Significance

To determine if the coin is fair, we compare the calculated probability (0.0439) with a significance level, typically 0.05. Since 0.0439 < 0.05, this result is statistically significant, suggesting that such an extreme outcome (8 tails and 2 heads) is unlikely under the assumption of a fair coin.

Step 4: Consider Alternative Hypotheses

Given the low probability under the fair coin assumption, we might consider alternative hypotheses such as:

• The coin is biased towards tails (p < 0.5 for heads).

Conducting further tests or experiments with a larger sample size can provide more robust conclusions.

Conclusion

Based on the statistical analysis, the observed outcome of 8 tails and 2 heads is unlikely to occur with a fair coin. While this suggests potential bias, further testing with a larger number of trials is recommended to make a definitive conclusion.