Ian and Sophia flip a fair coin ten times and record their results; they find that the coin landed on heads eight times. lan says, "This means the coin will land on heads \( 80 \% \) of the time!" Sophia says, "I don't think so; if we flip it many more times, our results should be closer to it landing on heads about \( 50 \% \) of the time." Who is right? How do you know? Choose the correct answer below. A. Ian is right, because 8 out of 10 is \( 80 \% \). B. lan is right, because a coin isn't neccessarily heads \( 50 \% \) of the time, and the sample probability is usually near the population probability. C. Sophia is right, because of the Law of Large Numbers (Bernoulli's Theorem). D. Sophia is right, because the coin should land on heads exactly \( 50 \% \) of the time.

**C. Sophia is right, because of the Law of Large Numbers (Bernoulli's Theorem).** **Explanation:** Ian and Sophia flipped a fair coin 10 times and observed 8 heads, which is an 80% outcome in this small sample. However, this small number of trials can deviate significantly from the true probability due to random chance. Sophia's perspective aligns with the **Law of Large Numbers**, which states that as the number of trials increases, the sample proportion (in this case, the percentage of heads) will converge to the expected probability (50% for a fair coin). While Ian's observation is correct for their specific set of 10 flips, Sophia is correct in asserting that with a much larger number of flips, the results should balance out closer to the true probability of 50%. **Why the other options are incorrect:** - **A.** While Ian correctly states that 8 out of 10 is 80%, he's incorrectly generalizing this small sample to the true probability of the coin. - **B.** This option is confusingly worded and incorrectly attributes Ian's correctness based on the coin not necessarily being fair, which contradicts the premise that the coin is fair. - **D.** Although Sophia is correct that the long-term probability approaches 50%, saying it should land "exactly" 50% of the time is misleading because individual trials can still vary. **Final Answer:** C. Sophia is right, because of the Law of Large Numbers (Bernoulli's Theorem).