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Chapter 8: Problem 32
The odds in favor of an event \(E\) occurring are 9 to 7 . What is the probability of \(E\) occurring?
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The number of average hours worked per year per worker in the United States and five European countries in 2002 is given in the following table: $$\begin{array}{lcccccc}\hline & & & \text { Great } & & \text { West } & \\\\\text { Country } & \text { U.S. } & \text { Spain } & \text { Britain } & \text { France } & \text { Germany } & \text {Norway } \\ \hline \text { Average } & & & & & & \\\\\text { Hours } & 1815 & 1807 & 1707 & 1545 & 1428 & 1342 \\ \text { Werked } & & & & & & \\ \hline\end{array}$$ Find the average of the average hours worked per worker in 2002 for workers in the six countries. What is the standard deviation for these data?
Roger Hunt intends to purchase one of two car dealerships currently for sale in a certain city. Records obtained from each of the two dealers reveal that their weekly volume of sales, with corresponding probabilities, are as follows: $$\begin{array}{l}\text { Dahl Motors }\\\ \begin{array}{lcccc}\hline \text { Cars Sold/Week } & 5 & 6 & 7 & 8 \\ \hline \text { Probability } & .05 & .09 & .14 & .24 \\\\\hline \end{array}\end{array}$$ $$\begin{array}{l}\text { Farthington Auto Sales }\\\ \begin{array}{lcccccc}\hline \text { Cars Sold/Week } & 5 & 6 & 7 & 8 & 9 & 10 \\\\\hline \text { Probability } & .08 & .21 & .31 & .24 & .10 & .06 \\\\\hline \end{array}\end{array}$$ The average profit/car at Dahl Motors is \(\$ 362\), and the average profit/car at Farthington Auto Sales is \(\$ 436\). a. Find the average number of cars sold each week at each dealership. b. If Roger's objective is to purchase the dealership that generates the higher weekly profit, which dealership should he purchase? (Compare the expected weekly profit for each dealership.)
In American roulette, as described in Example 6, a player may bet on a split (two adjacent numbers). In this case, if the player bets $$\$ 1$$ and either number comes up, the player wins $$\$ 17$$ and gets his $$\$ 1$$ back. If neither comes up, he loses his $$\$ 1$$ bet. Find the expected value of the winnings on a $$\$ 1$$ bet placed on a split.
Let \(X\) denote the random variable that gives the sum of the faces that fall uppermost when two fair dice are rolled. Find \(P(X=7)\).
a. Show that, for any number \(c\), $$E(c X)=c E(X)$$ b. Use this result to find the expected loss if a gambler bets \(\$ 300\) on \(\mathrm{red}\) in a single play in American roulette.
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