A student flips a coin 20 times and finds that the coin landed heads on 15 of the 20 flips. Does this result provide convincing evidence that the coin is not fair? In other words, does this result provide convincing evidence that the probability that the coin lands heads is not equal to the probability that the coin lands tails? Answer the following questions to investigate this research question. If a coin is fair and is flipped 20 times, approximately what fraction of the coin flips do you expect will land heads? Choose the best answer below.

If a coin is fair, the probability of it landing heads is \( \frac{1}{2} \), or 50%. Therefore, in 20 flips of a fair coin, we expect approximately 50% of them to land heads.

### To calculate the expected fraction:
\[
\text{Expected fraction of heads} = \frac{1}{2} = 0.5
\]
So, out of 20 flips, you would expect:
\[
\text{Expected number of heads} = 0.5 \times 20 = 10 \text{ heads}
\]
Thus, the fraction of heads out of 20 flips would be:
\[
\frac{10}{20} = 0.5
\]

### Best answer:
The expected fraction of heads is **0.5**.

Would you like to go into more detail on this result or the next part of the investigation? Here are some follow-up questions:

1. What is the next step to test if the coin is fair based on this result?
2. How do we calculate the variance or standard deviation of the number of heads?
3. What is the probability of getting exactly 15 heads out of 20 flips if the coin is fair?
4. Can we use a hypothesis test to determine if the coin is biased?
5. What is the significance level commonly used to test fairness in such experiments?

**Tip:** In hypothesis testing, calculating the standard deviation for binomial distributions is crucial to measure how far an outcome is from the expected value.

A student flips a coin 20 times and finds that the coin landed heads on 15 of the 20 flips. Does this result provide convincing evidence that the coin is not fair? In other words, does this result provide convincing evidence that the probability that the coin lands heads is not equal to the probability that the coin lands tails? If a coin is fair and is flipped 20 times, approximately what fraction of the coin flips do you expect will land heads?

Investigate whether a coin is fair by analyzing 20 flips, where the coin landed heads 15 times. Learn about probability, binomial distributions, and hypothesis testing. This problem explains how to calculate expected outcomes, variance, and perform statistical tests. Suitable for students in Grades 9-12.

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