91Ó°ÊÓ

A system is comprised of 5 components, each of which is either working or failed. Consider an experiment that consists of observing the status of each component, and let the outcome of the experiment be given by the vector \(\left(x_{1}, x_{2}, x_{3}, x_{4}, x_{5}\right)\) where \(x_{i}\) is equal to 1 if component \(i\) is working and is equal to 0 if component \(i\) is failed. (a) How many outcomes are in the sample space of this experiment? (b) Suppose that the system will work if components 1 and 2 are both working, or if components 3 and 4 are both working, or if components \(1,3,\) and 5 are all working. Let \(W\) be the event that the system will work. Specify all the outcomes in \(W\) (c) Let \(A\) be the event that components 4 and 5 are both failed. How many outcomes are contained in the event \(A ?\) (d) Write out all the outcomes in the event \(A W\).

If it is assumed that all \(\left(\begin{array}{c}52 \\ 5\end{array}\right)\) poker hands are equally likely, what is the probability of being dealt (a) a flush? (A hand is said to be a flush if all 5 cards are of the same suit. (b) one pair? (This occurs when the cards have denominations \(a, a, b, c, d,\) where \(a, b, c,\) and \(d\) are all distinct. (c) two pairs? (This occurs when the cards have denominations \(a, a, b, b, c,\) where \(a, b,\) and \(c\) are all distinct.) (d) three of a kind? (This occurs when the cards have denominations \(a, a, a, b, c,\) where \(a, b\) and \(c\) are all distinct.) (e) four of a kind? (This occurs when the cards have denominations \(a, a, a, a, b .)\)

A small community organization consists of 20 families, of which 4 have one child, 8 have two children, 5 have three children, 2 have four children, and 1 has five children. (a) If one of these families is chosen at random, what is the probability it has \(i\) children, \(i=\) \(1,2,3,4,5 ?\) (b) If one of the children is randomly chosen, what is the probability that child comes from a family having \(i\) children, \(i=1,2,3,4,5 ?\)

If two dice are rolled, what is the probability that the sum of the upturned faces equals \(i ?\) Find it for \(i=2,3, \ldots, 11,12\).

Five people, designated as \(A, B, C, D, E,\) are arranged in linear order. Assuming that each possible order is equally likely, what is the probability that (a) there is exactly one person between \(A\) and \(B ?\) (b) there are exactly two people between \(A\) and \(B\) ? (c) there are three people between \(A\) and \(B ?\)