Exam 8: Sets and Probability

Use a Venn diagram to answer the question. -A survey of a group of 112 tourists was taken in St. Louis. The survey showed the following: 63 of the tourists plan to visit Gateway Arch; 47 plan to visit the zoo; 9 plan to visit the Art Museum and the zoo, but not the Gateway Arch; 12 plan to visit the Art Museum and the Gateway Arch, but not the zoo; 18 plan to visit the Gateway Arch and the zoo, but not the Art Museum; 8 plan to visit the Art Museum, the zoo, and the Gateway Arch; 15 plan to visit none of the three places. How many plan to visit the Art Museum only?

Decide whether the statement is true or false. - $\{7,14,21,28\} \cup\{7,21\}=\{7,14,21,28\}$

Shade the Venn diagram to represent the set. - $C^{\prime} \cap(A \cup B)$

Find the probability. -If two cards are drawn with replacement from an ordinary deck, find the probability the first card is a heart and the second is a diamond.

Use a Venn diagram to answer the question. -A survey of 180 families showed that 67 had a dog; 52 had a cat; 22 had a dog and a cat; 70 had neither a cat nor a dog, and in addition did not have a parakeet; 4 had a cat, a dog, and a parakeet. How many had a parakeet only?

Solve the problem. Express the answer as a percentage. - $31 \%$ of the workers at Motor Works are female, while $37 \%$ of the workers at City Bank are female. If a worker is selected at random, what is the probability that the worker will be from Motor Works, given that the worker is female? Round your answer to the nearest tenth, if necessary.

Assume that two marbles are drawn without replacement from a box with 1 blue, 3 white, 2 green, and 2 red marbles. Find the probability of the indicated result. -The second marble is blue, given that the first marble is white.

Use a Venn diagram to decide if the statement is true or false. - $A \cap B^{\prime}=\left(A^{\prime} \cup B\right)^{\prime}$

Decide whether the two events listed are independent. -Two cards are selected, with replacement, from an ordinary deck. C is the event that a club appears on the first draw. A is the event that an ace appears on the second draw.

Write the sample space for the given experiment. -An 8 -sided die is rolled. (The sides contain the numbers $1,2,3,4,5,6,7$ , and 8 .)

Solve the problem, rounding the answer as appropriate. Assume that "pure dominant" describes one who has twodominant genes for a given trait; "pure recessive" describes one who has two recessive genes for a given trait; and"hybrid" describes one who has one of each. -In a population, $40 \%$ of the females are pure dominant, $50 \%$ are hybrid, and $10 \%$ are pure recessive. If a pure recessive male mates with a random female and their first offspring has the dominant trait, what is the probability that the female is hybrid?

Solve the problem using Bayes' Theorem. Round the answer to the nearest hundredth, if necessary. -For mutually exclusive events $X_{1}, X_{2}$ , and $X_{3}$ , let $P\left(X_{1}\right)=.19, P\left(X_{2}\right)=.27$ , and $P\left(X_{3}\right)=.54$ . Also, $\mathrm{P}\left(\mathrm{Y} \mid \mathrm{X}_{1}\right)=.40, \mathrm{P}\left(\mathrm{Y} \mid \mathrm{X}_{2}\right)=.30$ , and $\mathrm{P}\left(\mathrm{Y}_{\mid} \mathrm{X}_{3}\right)=.60$ . Find $\mathrm{P}\left(\mathrm{X}_{2} \mid \mathrm{Y}\right)$ .

Find the odds in favor of the indicated event. -Spinning a $D$ on the spinner pictured below. (The sectors are of equal size.)

Convert the odds that the given event will occur to the probability that the event will occur. -The odds in favor of becoming the President are 1 to $5,300,000$ .

Assume that, at a certain college, $33 \%$ of all physics majors belong to ethnic minorities. Given a random sample of 10physics majors, find the probability of the indicated event. Round your answer as appropriate. -Two or less belong to an ethnic minority.

For the experiment described, write the indicated event in set notation. -A coin is tossed three times. Represent the event "the first toss comes up tails" as a subset of the sample space.

Find the probability of the given event. -A card drawn from a well-shuffled deck of 52 cards is red.

The table shows, for some particular year, a listing of several income levels and, for each level, the proportion of the population in the level and the probability that a person in that level bought a new car during the year. Given that one of the people who bought a new car during that year is randomly selected, find the probability that that person was in the indicated income category. Round your answer to the nearest hundredth. - $\$ 15,000$ - $\$ 19,999$

Find the number of subsets of the set. -{8, 9, 10}

Use a Venn diagram to decide if the statement is true or false. - $A^{\prime} \cup(B \cap C)=\left(A^{\prime} \cup B\right) \cap\left(A^{\prime} \cup C\right)$