Chapter 14

The binomial coefficient \(\left(\begin{array}{l}6 \\ 4\end{array}\right)\) equals (a) \(\frac{6 !}{4 !}\) (b) \(\frac{6 !}{4 ! \cdot 2 !}\) (c) \(\frac{6 !}{2 !}\) (d) \(\frac{(6-4) !}{2 !}\)

True or False The intersection of two sets is always a subset of their union.

True or False. In a probability model, the sum of all probabilities is \(1 .\)

True or False If \(A\) is a set, the complement of \(A\) is the set of all the elements in the universal set that are not in \(A\).

In a probability model, which of the following numbers could be the probability of an outcome? $$\begin{array}{llllll}0 & 0.01 & 0.35 & -0.4 & 1 & 1.4\end{array}$$

In a probability model, which of the following numbers could be the probability of an outcome? $$\begin{array}{llll}1.5 & \frac{1}{2} & \frac{3}{4} & \frac{2}{3} & 0 & -\frac{1}{4}\end{array}$$

In Problems \(7-14\), find the value of each permutation. $$ P(6,2) $$

Determine whether the following is a probability model. $$\begin{array}{cc}\text { Outcome } & \text { Probability } \\\\\hline 1 & 0.2 \\\2 & 0.3 \\\3 & 0.1 \\\4 & 0.4\end{array}$$

True or False If a task consists of a sequence of three choices in which there are \(p\) selections for the first choice, \(q\) selections for the second choice, and \(r\) selections for the third choice, then the task of making these selections can be done in \(p \cdot q \cdot r\) different ways.

Find the value of each permutation. $$ P(7,2) $$

Determine whether the following is a probability model. $$\begin{array}{lc}\text { Outcome } & \text { Probability } \\\\\hline \text { Steve } & 0.4 \\\\\text { Bob } & 0.3 \\ \text { Faye } & 0.1 \\\\\text { Patricia } & 0.2\end{array}$$

Multiple Choice The Counting Formula states that if \(A\) and \(B\) are finite sets, then \(n(A \cup B)=\) __________. (a) \(n(A)+n(B)\) (b) \(n(A)+n(B)-n(A \cap B)\) (c) \(n(A) \cdot n(B)\) (d) \(n(A)-n(B)\)

Find the value of each permutation. $$ P(4,4) $$

Write down all the subsets of \(\\{a, b, c, d\\}\).

Find the value of each permutation. $$ P(8,8) $$

Determine whether the following is a probability model. $$\begin{array}{lc}\text { Outcome } & \text { Probability } \\\\\hline \text { Erica } & 0.3 \\\\\text { Joanne } & 0.2 \\\\\text { Laura } & 0.1 \\\\\text { Donna } & 0.5 \\\\\text { Angela } & -0.1\end{array}$$

Find the value of each permutation. $$ P(7,0) $$

List the sample space S of each experiment and (b) construct a probability model for the experiment. Tossing a fair coin twice

If \(n(A)=15, n(B)=20,\) and \(n(A \cap B)=10\) find \(n(A \cup B)\)

List the sample space S of each experiment and (b) construct a probability model for the experiment. Tossing two fair coins once

If \(n(A)=30, n(B)=40,\) and \(n(A \cup B)=45\) find \(n(A \cap B)\)

Find the value of each permutation. $$ P(8,4) $$

List the sample space S of each experiment and (b) construct a probability model for the experiment. Tossing two fair coins and then a fair die

If \(n(A \cup B)=50, n(A \cap B)=10,\) and \(n(B)=20\) find \(n(A) .\)

Find the value of each permutation. $$ P(8,3) $$

List the sample space S of each experiment and (b) construct a probability model for the experiment. Tossing a fair coin, a fair die, and then a fair coin

If \(n(A \cup B)=60, n(A \cap B)=40,\) and \(n(A)=n(B)\) find \(n(A) .\)

List the sample space S of each experiment and (b) construct a probability model for the experiment. Tossing three fair coins once

List the sample space S of each experiment and (b) construct a probability model for the experiment. Tossing one fair coin three times

List all the permutations of 5 objects \(a, b, c, d,\) and \(e\) choosing 3 at a time without repetition. What is \(P(5,3) ?\)

A man has 5 shirts and 3 ties. How many different shirt-and-tie arrangements can he wear?

List all the permutations of 5 objects \(a, b, c, d,\) and \(e\) choosing 2 at a time without repetition. What is \(P(5,2) ?\)

A woman has 5 blouses and 8 skirts How many different outfits can she wear?

List all the permutations of 4 objects \(1,2,3,\) and 4 choosing 3 at a time without repetition. What is \(P(4,3) ?\)

How many four-digit numbers can be formed using the digits \(0,1,2,3,4,5,6,7,8,\) and 9 if the first digit cannot be 0 ? Repeated digits are allowed.

List all the permutations of 6 objects \(1,2,3,4,5,\) and 6 choosing 3 at a time without repetition. What is \(P(6,3) ?\)

A coin is weighted so that heads is four times as likely as tails to occur. What probability should be assigned to heads? to tails?

List all the combinations of 5 objects \(a, b, c, d,\) and \(e\) taken 3 at a time. What is \(C(5,3) ?\)

In a consumer survey of 500 people, 200 indicated that they would be buying a major appliance within the next month, 150 indicated that they would buy a car, and 25 said that they would purchase both a major appliance and a car. How many will purchase neither? How many will purchase only a car?

A coin is weighted so that tails is twice as likely as heads to occur. What probability should be assigned to heads? to tails?

List all the combinations of 5 objects \(a, b, c, d,\) and \(e\) taken 2 at a time. What is \(C(5,2) ?\)

In a student survey, 200 indicated that they would attend Summer Session I, and 150 indicated Summer Session II. If 75 students plan to attend both summer sessions, and 275 indicated that they would attend neither session, how many students participated in the survey?

List all the combinations of 4 objects \(1,2,3,\) and 4 taken 3 at a time. What is \(C(4,3) ?\)

In a survey of 100 investors in the stock market, 50 owned shares in IBM 40 owned shares in AT\&T 45 owned shares in GE 20 owned shares in both IBM and GE 15 owned shares in both \(\mathrm{AT} \& \mathrm{~T}\) and \(\mathrm{GE}\) 20 owned shares in both IBM and AT\&T 5 owned shares in all three (a) How many of the investors surveyed did not have shares in any of the three companies? (b) How many owned just IBM shares? (c) How many owned just GE shares? (d) How many owned neither IBM nor GE? (e) How many owned either IBM or AT\&T but no GE?

List all the combinations of 6 objects \(1,2,3,4,5,\) and 6 taken 3 at a time. What is \(C(6,3) ?\)

Human blood is classified as either \(\mathrm{Rh}+\) or \(\mathrm{Rh}-.\) Blood is also classified by type: \(\mathrm{A},\) if it contains an A antigen but not a \(B\) antigen; \(B\), if it contains a \(\mathrm{B}\) antigen but not an \(\mathrm{A}\) antigen; \(\mathrm{AB},\) if it contains both \(\mathrm{A}\) and \(B\) antigens; and \(O\), if it contains neither antigen. Draw a Venn diagram illustrating the various blood types Based on this classification, how many different kinds of blood are there?

The sample space is \(S=\\{1,2,3,4,5,6,\) 7,8,9,10}. Suppose that the outcomes are equally likely. Compute the probability of the event \(E=\\{1,2,3\\}\).

How many two-letter codes can be formed using the letters \(A, B, C,\) and \(D ?\) Repeated letters are allowed.

How many two-letter codes can be formed using the letters \(A, B, C, D,\) and \(E\) ? Repeated letters are allowed.

Forming Codes How many two-letter codes can be formed using the letters \(A, B, C, D,\) and \(E ?\) Repeated letters are allowed.