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Identifying a Biased Coin
You have a fair coin and a biased coin, but you can’t tell which is which. The biased coin lands on heads 75% of the time. You decide to try to determine which coin is the biased coin by selecting one of the coins at random and flipping tn 100 times. Let \(\hat{p}\) be your observed fraction of heads. Based on \(\hat{p}\), you decide which coin is the biased one.
- For which values of \(\hat{p}\) will you assume the coin you flipped is the biased coin?
- What is the probability that you correctly determine which coin is the biased coin?
Biased coin
You have a biased coin, but you don’t know what the bias is. Let \(p\) be the actual probability of getting heads on a single coin flip, \(p=\mathbb{P}(Heads).\)
- Suppose \(p=0.8\). What is the probability of observing between 76 and 84 heads out of 100 flips of the coin.
- Suppose you flip the coin 100 times and observe 80 heads. What is the 95% confidence interval for \(p\)?
Random Digit
Let \(D_i\) be a random digit chosen uniformly from \(\{0,1,2,3,4,5,6,7,8,9\}\). Assume that each of the \(D_i\) are independent.
Let \(X_i\) be the last digit of \(D_i^2\). So if \(D_i=9\) then \(D_i^2=81\) and \(X_i=1\). Define \(\bar X_n\) by
\[\bar X_n = \frac{X_1 + \cdots+X_n}{n}\]
- Predict the value of \(\bar X_n \) when \(n\) is large.
- Find the number \(\epsilon\) such that for \(n=10,000\) the chance that you prediction is off by more than \(\epsilon\) is about 1/200.
- Find approximately the least value of \(n\) such that your prediction of \(\bar X_n\) is correct to within 0.01 with probability at least 0.99 .
- If you just had to predict the first digit of \(\bar X_{100}\), what digit should you choose to maximize your chance of being correct, and what is that chance ?
[Pitman p206, #30]
