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A deck of playing cards has $52$ cards, of which $12$ are face cards. If you shuffle the deck well and turn over the top $3$ cards, one after the other, what鈥檚 the probability that all 3 are face cards?$\left(a\right)0.001\left(b\right)0.005\left(c\right)0.010\left(d\right)0.012\left(e\right)0.02$

Your teacher has invented a 鈥渇air鈥� dice game to play. Here鈥檚 how it works. Your teacher will roll one fair eight-sided die, and you will roll a fair six-sided die. Each player rolls once, and the winner is the person with the higher number. In case of a tie, neither player wins. The table shows the sample space of this chance process.

(a) Let A be the event 鈥測our teacher wins.鈥� Find P(A).

(b) Let B be the event 鈥測ou get a $3$ on your first roll.鈥� Find P(A 鈭� B).

(c) Are events A and B independent? Justify your answer.

Three machines鈥擜, B, and C鈥攁re used to produce a large quantity of identical parts at a factory. Machine A produces $60\%$ of the parts, while Machines B and C produce $30\%$ and $10\%$ of the parts, respectively. Historical records indicate that $10\%$ of the parts produced by Machine A are defective, compared with $30\%$ for Machine B and $40\%$ for Machine C.

(a) Draw a tree diagram to represent this chance process.

(b) If we choose a part produced by one of these three machines, what鈥檚 the probability that it鈥檚 defective? Show your work.

(c) If a part is inspected and found to be defective, which machine is most likely to have produced it? Give appropriate evidence to support your answer.

Researchers are interested in the relationship between cigarette smoking and lung cancer. Suppose an adult male is randomly selected from a particular population. The following table shows the probabilities of some events related to this chance process:

(a) Find the probability that the individual gets cancer given that he is a smoker. Show your work.

(b) Find the probability that the individual smokes or gets cancer. Show your work.

(c) Two adult males are selected at random. Find the probability that at least one of the two gets cancer. Show your work.

Based on previous records,$17\%$ of the vehicles passing through a tollbooth have out-of-state plates. A bored tollbooth worker decides to pass the time by counting how many vehicles pass through until he sees two with out-of-state plates.$27$

(a) Describe the design of a simulation to estimate the average number of vehicles it takes to find two with out of- state plates. Explain clearly how you will use the partial table of random digits below to carry out your simulation.

(b) Perform three repetitions of the simulation you described in part (a). Copy the random digits below onto your paper. Then mark on or directly above the table to show your results.

$9$China has 1.2 billion people. Marketers want to know which international brands they have heard of. A large study showed that $62\%$ of all Chinese adults have heard of Coca-Cola. You want to simulate choosing a Chinese at

random and asking if he or she has heard of Coca-Cola. One correct way to assign random digits to simulate the answer is:

(a) One digit simulates one person鈥檚 answer; odd means 鈥淵es鈥� and even means 鈥淣o.鈥�

(b) One digit simulates one person鈥檚 answer; $0$to $6$mean 鈥淵es鈥� and $7$to $9$mean 鈥淣o. 鈥�

(c) One digit simulates the result; $0$to tells how many in the sample said 鈥淵es.鈥�

(d) Two digits simulate one person鈥檚 answer; $00$ to $61$ mean 鈥淵es鈥� and $62$ to $99$ mean 鈥淣o. 鈥�

(e) Two digits simulate one person鈥檚 answer; $00$ to $62$ mean 鈥淵es鈥� and $63$ to $99$ mean 鈥淣o. 鈥�

Choose an American household at random and record the number of vehicles they own. Here is the probability model if we ignore the few households that own more than $5$ cars: A housing company builds houses with two-car garages. What percent of households have more cars than the

garage can hold? $\left(a\right)7\%\left(b\right)13\%\left(c\right)20\%\left(d\right)45\%\left(e\right)55\%$

Computer voice recognition software is getting better. Some companies claim that their software correctly recognizes $98\%$of all words spoken by a trained user. To simulate recognizing a single word when the probability of being correct is $0.98$, let two digits simulate one word; $00$ to $97$ mean 鈥渃orrect.鈥� The program recognizes words (or not) independently. To simulate the program鈥檚 performance on $10$ words, use these random digits:

$6097070024178682984361790906568796418883$ The number of words recognized correctly out of the $10$ is $\left(a\right)10\left(b\right)9\left(c\right)8\left(d\right)7\left(e\right)6$

What is the probability that a student has a GPA under $2.0$ or has skipped many classes? $\left(a\right)0.080\left(b\right)0.281\left(c\right)0.285\left(d\right)0.365\left(e\right)0.727$

What is the probability that a student has a GPA under$2.0$ given that he or she has skipped many classes? $\left(a\right)0.080\left(b\right)0.281\left(c\right)0.285\left(d\right)0.314\left(e\right)0.727$