Describe the sample space \(S\) of the experiment, and list the elements of the given event. (Assume that the coins are distinguishable and that what is observed are the faces or numbers that face up.) Two coins are tossed; the result is at most one tail.

Complete the following. The relative frequency of an event \(E\) is defined to be __________

Complete the following sentence. An event is a _____.

In fall 2002, UCLA admitted \(26 \%\) of its California resident applicants, $18 \%\( of its applicants from other U.S. states, and \)13 \%$ of its international student applicants. Of all its applicants, \(86 \%\) were California residents, \(11 \%\) were from other U.S. states, and \(3 \%\) were international students. \({ }^{78}\) What percentage of all admitted students were California residents? (Round your answer to the nearest \(1 \% .)\)

Are based on the following table, which shows the stock market performance of 40 industries from five sectors of the U.S. economy as of noon on September \(11,2015 .\) (Take S to be the set of all 40 industries represented in the table.) $$\begin{array}{|r|c|c|c|c|}\hline & \begin{array}{c}\text { Increased } \\\\(\boldsymbol{X})\end{array} & \begin{array}{c}\text { Decreased } \\\\(\boldsymbol{Y})\end{array} & \begin{array}{c}\text { Unchanged } \\\\(\boldsymbol{Z})\end{array} & \text { Totals } \\\\\hline \text { Financials }(\boldsymbol{F}) & 3 & 4 & 1 & 8 \\\\\hline \text { Manufacturing }(\boldsymbol{M}) & 8 & 3 & 3 & 14 \\\\\hline \begin{array}{r}\text { Information } \\\\\text { Technology }(T)\end{array} & 6 & 1 & 0 & 7\\\\\hline \text { Health Care }(\boldsymbol{H}) & 4 & 1 & 1 & 6 \\\\\hline \text { Utilities }(\boldsymbol{U}) & 3 & 1 & 1 & 5 \\\\\hline \text { Totals } & 24 & 10 & 6 & 40 \\\\\hline\end{array}$$ Use symbols to describe the event that an industry was in the manufacturing sector and did not increase in value. How many elements are in this event?

Write down an expanded form of Bayes' theorem that applies to a partition of the sample space \(S\) into four events \(R_{1}, R_{2}, R_{3},\) and \(R_{4}\).