As the number of coin flips increases, the proportion of heads should become closer to 50%.

Question

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The statement presents a classic idea from probability and statistics about coin flips.

Consider a fair coin, where the probability of heads on each flip is 0.5 and flips are independent. If we look at the proportion of heads after n flips, this proportion is a random variable that follows a binomial distribution with parameters n and p = 0.5, divided by n. As n grows larger, the law of large numbers tells us that this sample proportion converges to the true probability p = 0.5 almost surely. Intuitively, more flips provide more data, so the random fluctuations average out and the observed proportion gets closer to 50%.

The standard error of the sample proportion is sqrt[p(1-p)/n]. For p = 0.5, this becomes sqrt[0.25/n] = 0.5 / sqrt(n). As n increases, this standard error shrinks, meaning the distribution of observed proportions becomes more tightly clustered around 0.5. This illustrates why the proportion of heads is expected to approach 50% with more flips.

Potential pitfalls or misconceptions:
- If the coin is not fair and has a different head probability p ≠ 0.5, the long-run proportion will approach p, not 0.5.
- For small numbers of flips, the proportion can deviate substantially from 0.5 due to randomness, so the statement is not about small n but about increasing n.
- The result speaks to the expected behavior over repeated experiments; individual sequences can still show deviations, but the overall tendency as n grows is toward 0.5.

In summary, assuming a fair coin and independent flips, the proportion of heads indeed becomes closer to 50% as the number of flips increases, reflecting convergence to p = 0.5.

STAT 1450.02 SP2025 (28132) Checkpoint 12.1 & 12.2

As the number of coin flips increases, the proportion of heads should become closer to 50%.

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STAT 1450.02 SP2025 (28132) Checkpoint 12.1 & 12.2

As the number of coin flips increases, the proportion of heads should become closer to 50%.

View Explanation

Log in for full answers

Similar Questions

Select all statements that are correct. The law of large numbers and central limit theorem (taken together) imply that:

As the sample size increases, the expected value of noise Stays the Same

Which of the following statements is true about the Law of Large Numbers?

Consider two pairs of grandparents. The first pair has 4 grandchildren and the second pair has 31 grandchildren. Which of the two pairs is more likely to have between​ 40% and​ 60% boys as​ grandchildren, assuming that boys and girls are equally likely as​ children? Why?

__________________ refers to the products, methods, inventions, and standards used for the purpose of producing information.

____________________ is data presented in a meaningful context.

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Consider two pairs of grandparents. The first pair has 4 grandchildren and the second pair has 31 grandchildren. Which of the two pairs is more likely to have between 40% and 60% boys as grandchildren, assuming that boys and girls are equally likely as children? Why?