Exercise · ZW-LAW-OF-LARGE-NUMBERS-001 · ★☆☆ Basic
A fair coin is tossed many times. What value does the proportion of heads approach as the number of tosses grows?
Choose one option.
Show answer
- B. \(\tfrac12\)
Explanation
The proportion of heads is the average of Bernoulli variables with expectation \(\tfrac12\); by the law of large numbers it tends to that value.
Review the concept: Law of large numbers
The law of large numbers states that the average \(\bar X_n\) of \(n\) independent observations of a variable \(X\) approaches its expectation \(\mu\) as \(n\) grows.
\[\bar X_n=\dfrac1n\sum_{i=1}^{n}X_i\ \xrightarrow[n\to\infty]{}\ \mu\]
More practice problems: law of large numbers
- ZW-LAW-OF-LARGE-NUMBERS-002 The variables X_i are independent with variance σ^2=25. What is the variance of bar X_n with n=100?
- ZW-LAW-OF-LARGE-NUMBERS-003 With σ^2=1 and n=100, what bound does Chebyshev give for P(|bar X_n-μ|≥0.5)?
- ZW-LAW-OF-LARGE-NUMBERS-004 With σ^2=4, what is the smallest n such that Var(bar X_n)≤0.01?
- ZW-LAW-OF-LARGE-NUMBERS-005 What is the expected value of bar X_n if each X_i has expectation μ?
- ZW-LAW-OF-LARGE-NUMBERS-006 A fair die is rolled many times. What value does the average of the results tend to?
- ZW-LAW-OF-LARGE-NUMBERS-007 With σ^2=9 and n=1000, what bound does Chebyshev give for P(|bar X_n-μ|≥0.3)?
- ZW-LAW-OF-LARGE-NUMBERS-008 A fair coin came up heads 5 times in a row. What is the probability of heads on the next toss?
- ZW-LAW-OF-LARGE-NUMBERS-009 Let S_n be the number of heads in n tosses of a fair coin. Which statement does the law of large nu…