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Mammograms Many women choose to have annual mammograms to screen for breast cancer after age 40\. A mammogram isn't foolproof. Sometimes the test suggests that a woman has breast cancer when she really doesn't (a "false positive"). Other times the test says that a woman doesn't have breast cancer when she actually does (a "false negative"). Suppose the false negative rate for a mammogram is \(0.10 .\) (a) Interpret this probability as a long-nun relative frequency. (b) Which is a more serious error in this case: a false positive or a false negative? Justify your answer.

Texas hold 'em In the popular Texas hold 'em variety of poker, players make their best five-card poker hand by combining the two cards they are dealt with three of five cards available to all players. You read in a book on poker that if you hold a pair (two cards of the same rank) in your hand, the probability of getting four of a kind is \(88 / 1000\). (a) Explain what this probability means. (b) If you play 1000 such hands, will you get four of a kind in exactly 88 of them? Explain.

Playing "Pick \(4 "\) The Pick 4 games in many state lotteries announce a four- digit winning number each day. You can think of the winning number as a four- digit group from a table of random digits. You win (or share) the jackpot if your choice matches the winning number. The winnings are divided among all players who matched the winning number. That suggests a way to get an edge. (a) The winning number might be, for example, either 2873 or \(9999 .\) Explain why these two outcomes have exactly the same probability. (b) If you asked many people whether 2873 or 9999 is more likely to be the randomly chosen winning number, most would favor one of them. Use the information in this section to say which one and to explain why. How might this affect the four-digit number you would choose?

Ten percent of U.S. households contain 5 or more people. You want to simulate choosing a household at random and recording "Yes" if it contains 5 or more people. Which of these are correct assignments of digits for this simulation? (a) Odd \(=\) Yes; Even \(=\) No (b) \(0=\) Yes; \(1-9=\) No (c) \(0-5=\) Yes; \(6-9=\) No (d) \(0-4=\) Yes; \(5-9=\) No (e) None of these

Role-playing games Computer games in which the players take the roles of characters are very popular. They go back to earlier tabletop games such as Dungeons \(\&\) Dragons. These games use many different types of dice. A four- sided die has faces with \(1,2,3,\) and 4 spots. (a) List the sample space for rolling the die twice (spots showing on first and second rolls). (b) What is the assignment of probabilities to outcomes in this sample space? Assume that the die is perfectly balanced.

Blood types All human blood can be typed as one of \(\mathrm{O}, \mathrm{A}, \mathrm{B},\) or \(\mathrm{AB},\) but the distribution of the types varies a bit with race. Here is the distribution of the blood type of a randomly chosen black American: $$\begin{array}{lcccc}\hline \text { Blood type: } & 0 & \mathrm{~A} & \mathrm{~B} & \mathrm{AB} \\\\\text { Probability: } & 0.49 & 0.27 & 0.20 & ? \\\\\hline\end{array}$$ (a) What is the probability of type \(A B\) blood? Why? (b) What is the probability that the person chosen does not have type \(A B\) blood? (c) Maria has type \(\mathrm{B}\) blood. She can safely receive blood transfusions from people with blood types \(\mathrm{O}\) and \(\mathrm{B}\). What is the probability that a randomly chosen black American can donate blood to Maria?

Facebook versus YouTube A recent survey suggests that \(85 \%\) of college students have posted a profile on Facebook, \(73 \%\) use YouTube regularly, and \(66 \%\) do both. Suppose we select a college student at random. (a) Make a two-way table for this chance process. (b) Construct a Venn diagram to represent this setting. (c) Consider the event that the randomly selected college student has posted a profile on Facebook or uses YouTube regularly. Write this event in symbolic form based on your Venn diagram in part (b). (d) Find the probability of the event described in part (c). Explain your method.

Foreign-language study Choose a student in grades 9 to 12 at random and ask if he or she is studying a language other than English. Here is the distribution of results: $$\begin{array}{lccccc}\hline \text { Language: } & \text { Spanish } & \text { French } & \text { German } & \text { All others } & \text { None } \\\\\text { Probability: } & 0.26 & 0.09 & 0.03 & 0.03 & 0.59 \\\\\hline\end{array}$$ (a) What's the probability that the student is studying a language other than English? (b) What is the conditional probability that a student is studying Spanish given that he or she is studying some language other than English?

Facebook versus YouTube A recent survey suggests that \(85 \%\) of college students have posted a profile on Facebook, \(73 \%\) use YouTube regularly, and \(66 \%\) do both. Suppose we select a college student at random and learn that the student has a profile on Facebook. Find the probability that the student uses YouTube regularly. Show your work.

Testing the test Are false positives too common in some medical tests? Researchers conducted an experiment involving 250 patients with a medical condition and 750 other patients who did not have the medical condition. The medical technicians who were reading the test results were unaware that they were subjects in an experiment. (a) Technicians correctly identified 240 of the 250 patients with the condition. They also identified 50 of the healthy patients as having the condition. What were the false positive and false negative rates for the test? (b) Given that a patient got a positive test result, what is the probability that the patient actually had the medical condition? Show your work.