Category Archives: Confidence Interval

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.

  1. For which values of \(\hat{p}\) will you assume the coin you flipped is the biased coin?
  2. 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, \(p=\mathbb{P}(Heads).\)

  1. Suppose \(p=0.8\). What is the probability of observing between 76 and 84 heads out of 100 flips of the coin.
  2. 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

\[\bar X_n = \frac{X_1 + \cdots+X_n}{n}\]

  1. Predict the value of \(\bar X_n \) when \(n\) is large.
  2. 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.
  3. 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 .
  4. 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]