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Drug testing Athletes are often tested for use of performance-enhancing drugs. Drug tests aren鈥檛 perfect鈥攖hey sometimes say that an athlete took a banned substance when that isn鈥檛 the case (a 鈥渇alse positive鈥�). Other times, the test concludes that the athlete is 鈥渃lean鈥� when he or she actually took a banned substance (a 鈥渇alse negative鈥�). For one commonly used drug test, the probability of a false negative is $0.03$.

(a) Interpret this probability as a long-run relative frequency.

(b) Which is a more serious error in this case: a false positive or a false negative? Justify your answer.

Scrabble In the game of Scrabble, each player begins by drawing 7 tiles from a bag containing 100 files. There are 42 vowels, 56 consonants, and 2 blank tiles in the bag. Chait chooses her 7 tiles and is surprised to discover that all of them are vowels. We can use a simulation to see if this result is likely to happen

by chance.

(a) State the question of interest using the language of probability.

(b) How would you use random digits to imitate one repetition of the process? What variable would you measure?

(c) Use the line of random digits below to perform one repetition. Copy these digits onto your paper. Mark directly on or above them to show how you

determined the outcomes of the change process. 00694 05977 19664 65441 20903 62371 22725 53340

(d) In 1000 repetitions of the simulation, there were 2 times when all 7 tiles were vowels. What conclusion would you draw?

Refer to the golden ticket parking lottery example. At the following month鈥檚 school assembly, the two lucky winners were once again members of the AP Statistics class. This raised suspicions about how the lottery was being conducted. How would you modify thesimulation in the example to estimate the probability of this happening just by chance?

Shuffle a standard deck of cards, and turn over the top card. Put it back in the deck, shuffle again, and turn over the top card. Define events $A$: first card is a heart, and $B$: second card is a heart.

Make a two-way table that displays the sample space.

The birthday problem What鈥檚 the probability that in a randomly selected group of$30$ unrelated people, at least two have the same birthday? Let鈥檚 make two

assumptions to simplify the problem. First, we鈥檒l ignore the possibility of a February $29$ birthday. Second, we assume that a randomly chosen person is equally likely to be born on each of the remaining $365$ days of the year.

(a) How would you use random digits to imitate one repetition of the process? What variable would you measure?

(b) Use technology to perform $5$ repetitions. Record the outcome of each repetition.

(c) Would you be surprised to learn that the theoretical probability is $0.71$? Why or why not?

Shuffle a standard deck of cards, and turn over the top two cards, one at a time. Define events $A$: first card is a heart, and $B$: second card is a heart.

Find $P(AandB)$.

Refer to the NASCAR and breakfast cereal example. What if the cereal company decided to make it harder to get some drivers鈥� cards than others? For instance, suppose the chance that each card appears in a box of the cereal is Jeff Gordon, $10\%$; Dale Earnhardt, Jr., $30\%$; Tony Stewart, $20\%$; Danica Patrick, $25\%$; and Jimmie Johnson,$15\%$. How would you modify the simulation in the example to estimate the chance that a fan would have to buy $23$ or more boxes to get the full set?

Monty Hall problem In Parade magazine, a reader posed the following question to Marilyn vos Savant and the 鈥淎sk Marilyn鈥� column: Suppose you鈥檙e on a game show, and you鈥檙e given the choice of three doors. Behind one door is a car, behind the others, goats. You pick a door, say #1, and the host, who knows what鈥檚 behind the doors, opens another door, say #3, which has a

goat. He says to you, 鈥淒o you want to pick door #2?鈥� Is it to your advantage to switch your choice of doors? The game show in question was Let鈥檚 Make a Deal and the host was Monty Hall. Here鈥檚 the first part of Marilyn鈥檚 response: 鈥淵es; you should switch. The first door has a $1/3$ chance of winning, but

the second door has a $2/3$ chance.鈥� Thousands of readers wrote to Marilyn to disagree with her answer. But she held her ground.

(a) Use an online Let鈥檚 Make a Deal applet to perform at least $50$ repetitions of the simulation. Record whether you stay or switch (try to do each about half

the time) and the outcome of each repetition.

(b) Do you agree with Marilyn or her readers? Explain.