91Ó°ÊÓ

Suppose that we want to generate the outcome of the flip of a fair coin but that all we have at our disposal is a biased coin which lands on heads with some unknown probability \(p\) that need not be equal to \(\frac{1}{2}\). Consider the following procedure for accomplishing our task. 1\. Flip the coin. 2\. Flip the coin again. 3\. If both fli?s land heads or both land tails, retum to step 1 . 4\. Let the result of the last flip be the result of the experiment. (a) Show that the result is equally likely to be either heads or tails. (b) Could we use a simpler procedure that continues to flip the coin until the last two flips are different and then lets the result be the outcome of the final flip?

Independent flips of a coin that lands on heads with probability \(p\) are made. What is the probability that the first four outcomes are (a) \(H, H, H, H ;\) (b) \(T, H, H, H ?\) (c) What is the probability that the pattern \(T, H, H, H\) occurs before the pattern \(H, H, H, H\) ?

An engineering system consisting of \(n\) components is said to be a \(k\)-out- of\(n\) system \((k \leq n)\) if the system functions if and only if at least \(k\) of the \(n\) components function. Suppose that all components function independently of each other. (a) If the \(i\) th component functions with probability \(P_{t}, i=1,2,3,4\), compute the probability that a 2-out-of-4 system functions. (b) Repeat part (a) for a 3-out-of-5 system. (c) Repeat for a \(k\)-out-of- \(n\) system when all the \(P_{i}\) equal \(p\) (that is, \(P_{i}=p\), \(i=1,2, \ldots, n)\)

There is a 50 - 50 chance that the queen carries the gene for hemophilia. If she is a carrier, then each prince has a 50-50 chance of having hemophilia. If the queen has had three princes without the disease, what is the probability the queen is a carrier? If there is a fourth prince, what is the probability that he will have hemophilia?

Let \(S=\\{1,2, \ldots, n\\}\) and suppose that \(A\) and \(B\) are, independently, equally likely to be any of the \(2^{n}\) subsets (including the null set and \(S\) itself) of \(S\). (a) Show that $$ P\\{A \subset B\\}=\left(\frac{3}{4}\right)^{n} $$ HINT: Let \(N(B)\) denote the number of elements in \(B\). Use $$ P\\{A \subset B\\}=\sum_{t=0}^{n} P\\{A \subset B \mid N(B)=i\\} P\\{N(B)=i\\} $$ (b) Show that \(P\\{A B=\varnothing\\}=\left(\frac{3}{4}\right)^{n}\).