Exam 8: Sets and Probability

Use the given table to find the indicated probability. -Co llege students were given three choices of pizza toppings and asked to choose one favorite. The following table shows the results. $\mathrm{P}$ (favorite topping is meat | student is junior)? Round the answer to the nearest hundredth.

Let $U=\{q, r, s, t, u, v, w, x, y, z\} ; A=\{q, s, u, w, y\} ; B=\{q, s, y, z\} ;$ and $C=\{v, w, x, y, z\}$ . List the members of the indicated set, using set braces. - $\mathrm{C}^{\prime} \cup \mathrm{A}^{\prime}$

Decide whether the statement is true or false. - $\{4,8,12,16\} \cap\{4,12\}=\{4,8,12,16\}$

You are being dealt a card from a standard deck. If you happen to see the bottom card before you are dealt the top card, will the probability that you are dealt a king still be $\frac{1}{13}$ ?

Shade the Venn diagram to represent the set. - $\mathrm{A}^{\prime} \cup \mathrm{B}^{\prime}$

Use the given table to find the probability of the indicated event. Round your answer to the nearest thousandth. -Co llege students were given three choices of pizza toppings and asked to choose one favorite. The following table shows the results. A randomly selected student prefers a cheese topping.

Solve the problem using Bayes' Theorem. Round the answer to the nearest hundredth, if necessary. -For two events $M$ and $N, P(M)=.3, P(N \mid M)=.7$ , and $P\left(N \mid M^{\prime}\right)=.9$ . Find $P(M \mid N)$ .

A die is rolled twice. Write the indicated event in set notation. -The sum of the rolls is either 3 or 4.

Provide an appropriate response. -In using the following form of Bayes' theorem, $\mathrm{P}(\mathrm{R} \mid \mathrm{T})=\frac{\mathrm{P}(\mathrm{T} \mid \mathrm{R}) \mathrm{P}(\mathrm{R})}{\mathrm{P}(\mathrm{T} \mid \mathrm{R}) \mathrm{P}(\mathrm{R})+\mathrm{P}\left(\mathrm{T} \mid \mathrm{R}^{\prime}\right) \mathrm{P}\left(\mathrm{R}^{\prime}\right)}$ , can $\mathrm{P}(\mathrm{T})$ , if it is known, be substituted for the entire denominator?

Find the probability of the given event. -A card drawn from a well-shuffled deck of 52 cards is a face card or a 6.

Provide an appropriate response. -Given that a student correctly uses the following form of Bayes' theorem, $\mathrm{P}\left(\mathrm{R}_{1} \mid \mathrm{T}\right)=\frac{\mathrm{P}\left(\mathrm{T} \mid \mathrm{R}_{1}\right) \mathrm{P}\left(\mathrm{R}_{1}\right)}{\mathrm{P}\left(\mathrm{T} \mid \mathrm{R}_{1}\right) \mathrm{P}\left(\mathrm{R}_{1}\right)+\mathrm{P}\left(\mathrm{T} \mid \mathrm{R}_{2}\right) \mathrm{P}\left(\mathrm{R}_{2}\right)+\mathrm{P}\left(\mathrm{T} \mid \mathrm{R}_{3}\right) \mathrm{P}\left(\mathrm{R}_{3}\right)}$ , Make a precise statement about the value of $P\left(R_{1} \cap R_{2} \cap R_{3}\right)$ .

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

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. -Two hybrids produce a litter of four offspring. What is the probability that only the first one is pure recessive?

If $P(A \cap B)=0.2$ , can $P(A)=0.7, P(B)=0.7$ , and $A$ and $B$ be independent events?

Find the odds in favor of the indicated event. -Sp inning an A on the spinner pictured below. (The sectors are of equal size.)

Use a Venn Diagram and the given information to determine the number of elements in the indicated set. - $n(U)=114, n(A)=44, n(B)=39, n(C)=48, n(A \cap B)=5, n(A \cap C)=8, n(B \cap C)=8$ , and $n(A \cap(B \cap C))=4$ . Find $n\left(A \cap(B \cup C)^{\prime}\right)$ .

Let $U=\{q, r, s, t, u, v, w, x, y, z\} ; A=\{q, s, u, w, y\} ; B=\{q, s, y, z\} ;$ and $C=\{v, w, x, y, z\}$ . List the members of the indicated set, using set braces. - $(A \cap B)^{\prime}$

Insert " $\subseteq$ " or "q" in the blank to make the statement true. - $\{4,6,8\} \_\{3,4,5,6,8\}$

Find the probability of the given event. -A bag contains 13 balls numbered 1 through 13. A randomly chosen ball has an even number.

Find the number of subsets of the set. - $\{x \mid x$ is a day of the week $\}$