Chapter 7

A pair of dice is rolled, and the number that appears uppermost on each die is observed. Refer to this experiment and find the probability of the given event. The sum of the numbers is an even number.

List the simple events associated with each experiment. A nickel and a dime are tossed, and the result of heads on tails is recorded for each coin.

Let \(S=\\{a, b, c, d, e, f\\}\) be a sample space of an experiment and let \(E=\\{a, b\\}, F=\\{a, d, f\\}\), and \(G=\\{b, c, e\\}\) be events of this experiment. Find the events \(E \cup F\) and \(E \cap F\).

Evaluate the given expression. $$ 3 \cdot 5 ! $$

Lynbrook West, an apartment complex financed by the State Housing Finance Agency, consists of one-, two-, three-, and four-bedroom units. The rental rate for each type of unit-low, moderate, or market-is determined by the income of the tenant. How many different rates are there?

Verify the equation $$ n(A \cup B)=n(A)+n(B) $$ for the given disjoint sets. $$ A=\\{a, e, i, o, u\\} \text { and } B=\\{g, h, k, l, m\\} $$

Write the set in set-builder notation. The set of gold medalists in the 2010 Winter Olympic Games

A pair of dice is rolled, and the number that appears uppermost on each die is observed. Refer to this experiment and find the probability of the given event. The sum of the numbers is either 7 or 11 .

List the simple events associated with each experiment. A card is selected at random from a standard 52 -card deck, and its suit- hearts \((h)\), diamonds \((d)\), spades \((s)\), or clubs (c) - is recorded.

Let \(S=\\{a, b, c, d, e, f\\}\) be a sample space of an experiment and let \(E=\\{a, b\\}, F=\\{a, d, f\\}\), and \(G=\\{b, c, e\\}\) be events of this experiment. Find the events \(F \cup G\) and \(F \cap G\).

Evaluate the given expression. $$ 2 \cdot 7 ! $$

Five different types of monthly commuter passes are offered by a city's local transit authority for each of three different groups of passengers: youths, adults, and senior citizens. How many different kinds of passes must be printed each month?

Verify the equation $$ n(A \cup B)=n(A)+n(B) $$ for the given disjoint sets. \(A=\\{x \mid x\) is a whole number between 0 and 4\(\\}\) \(B=\\{x \mid x\) is a negative integer greater than \(-4\\}\)

A pair of dice is rolled, and the number that appears uppermost on each die is observed. Refer to this experiment and find the probability of the given event. A pair of 1 s is thrown.

List the simple events associated with each experiment. An opinion poll is conducted among a group of registered voters. Their political affiliationDemocrat (D), Republican ( \(R\) ), or Independent \((I)\) -and their sex-male \((m)\) or female \((f)-\) are recorded.

Let \(S=\\{a, b, c, d, e, f\\}\) be a sample space of an experiment and let \(E=\\{a, b\\}, F=\\{a, d, f\\}\), and \(G=\\{b, c, e\\}\) be events of this experiment. Find the events \(F^{c}\) and \(E \cap G^{c}\).

Evaluate the given expression. $$ \frac{5 !}{2 ! 3 !} $$

In the game of blackjack, a 2-card hand consisting of an ace and either a face card or a 10 is called a "blackjack." If a standard 52-card deck is used, determine how many blackjack hands can be dealt. (A "face card" is a jack, queen, or king.)

Let \(A=\\{2,4,6,8\\}\) and \(B=\\{6,7,8,9,10\\} .\) Compute: a. \(n(A)\) b. \(n(B)\) c. \(n(A \cup B)\) d. \(n(A \cap B)\)

Write the set in set-builder notation. $$ \\{3,4,5,6,7\\} $$

A pair of dice is rolled, and the number that appears uppermost on each die is observed. Refer to this experiment and find the probability of the given event. A double is thrown.

List the simple events associated with each experiment. As part of a quality-control procedure, eight circuit boards are checked, and the number of defective boards is recorded.

Let \(S=\\{a, b, c, d, e, f\\}\) be a sample space of an experiment and let \(E=\\{a, b\\}, F=\\{a, d, f\\}\), and \(G=\\{b, c, e\\}\) be events of this experiment. Find the events \(E^{c}\) and \(F^{c} \cap G\).

Evaluate the given expression. $$ \frac{6 !}{4 ! 2 !} $$

A coin is tossed 4 times and the sequence of heads and tails is recorded. a. Use the generalized multiplication principle to determine the number of outcomes of this activity. b. Exhibit all the sequences by means of a tree diagram.

Let \(U=\\{1,2,3,4,5,6,7, a, b, c, d, e\\} .\) If \(A=\\{1,2, a, e\\}\) and \(B=\\{1,2,3,4, a, b, c\\}\), find: a. \(n\left(A^{c}\right)\) b. \(n\left(A \cap B^{c}\right)\) c. \(n\left(A \cup B^{c}\right)\) d. \(n\left(A^{c} \cap B^{c}\right)\)

Write the set in set-builder notation. $$ \\{1,3,5,7,9,11, \ldots, 39\\} $$

A pair of dice is rolled, and the number that appears uppermost on each die is observed. Refer to this experiment and find the probability of the given event. One die shows a 6 , and the other is a number less than 3 .

List the simple events associated with each experiment. In a survey conducted to determine whether movie attendance is increasing \((i)\), decreasing \((d)\), or holding steady \((s)\) among various sectors of the population, participants are classified as follows: Group 1: Those aged 10-19 Group 2: Those aged 20-29 Group 3: Those aged 30-39 Group 4: Those aged 40-49 Group 5: Those aged 50 and older The response and age group of each participant are recorded.

Let \(S=\\{a, b, c, d, e, f\\}\) be a sample space of an experiment and let \(E=\\{a, b\\}, F=\\{a, d, f\\}\), and \(G=\\{b, c, e\\}\) be events of this experiment. Are the events \(E\) and \(F\) mutually exclusive?

Evaluate the given expression. $$ P(5,5) $$

A female executive selecting her wardrobe purchased two blazers, four blouses, and three skirts in coordinating colors. How many ensembles consisting of a blazer, a blouse, and a skirt can she create from this collection?

Verify directly that \(n(A \cup B)=n(A)+n(B)-n(A \cap B)\) for the sets in Exercise \(3 .\)

List the elements of the set in roster notation. $$ \\{x \mid x \text { is a digit in the number } 352,646\\} $$

A pair of dice is rolled, and the number that appears uppermost on each die is observed. Refer to this experiment and find the probability of the given event. The sum of the numbers is at least 4 .

List the simple events associated with each experiment. Data concerning durable goods orders are obtained each month by an economist. A record is kept for a 1 -yr period of any increase \((i)\), decrease \((d)\), or unchanged movement \((u)\) in the number of durable goods orders for each month as compared with the number of such orders in the same month of the previous year.

Let \(S=\\{a, b, c, d, e, f\\}\) be a sample space of an experiment and let \(E=\\{a, b\\}, F=\\{a, d, f\\}\), and \(G=\\{b, c, e\\}\) be events of this experiment. Are the events \(E \cup F\) and \(E \cap F^{c}\) mutually exclusive?

Evaluate the given expression. $$ P(6,6) $$

Four commuter trains and three express buses depart from city A to city B in the morning, and three commuter trains and three express buses operate on the return trip in the evening. In how many ways can a commuter from city A to city B complete a daily round trip via bus and/or train?

Let \(A=\\{a, e, i, o, u\\}\) and \(B=\\{b, d, e, o, u\\}\). Verify by direct computation that \(n(A \cup B)=n(A)+n(B)-n(A \cap B)\).

List the elements of the set in roster notation. $$ \\{x \mid x \text { is a letter in the word HIPPOPOTAMUS }\\} $$

An experiment consists of selecting a card at random from a 52-card deck. Refer to this experiment and find the probability of the event. A king of diamonds is drawn.

List the simple events associated with each experiment. Blood tests are given as a part of the admission procedure at the Monterey Garden Community Hospital. The blood type of each patient (A, B, AB, or O) and the presence or absence of the Rh factor in each patient's blood \(\left(\mathrm{Rh}^{+}\right.\) or \(\mathrm{Rh}^{-}\) ) are recorded.

Let \(S=\\{1,2,3,4,5,6\\}, E=\\{2,4,6\\}\) \(F=\\{1,3,5\\}\), and \(G=\\{5,6\\}\). Find the event \(E \cup F \cup G\).

Evaluate the given expression. $$ P(5,2) $$

If \(n(A)=15, n(A \cap B)=5\), and \(n(A \cup B)=30\), then what is \(n(B)\) ?

List the elements of the set in roster notation. $$ \\{x \mid 2-x=4 \text { and } x \text { is an integer }\\} $$

An experiment consists of selecting a card at random from a 52-card deck. Refer to this experiment and find the probability of the event. A diamond or a king is drawn.

List the simple events associated with each experiment. A meteorologist preparing a weather map classifies the expected average temperature in each of five neighboring states (MN, WI, IA, IL, MO) for the upcoming week as follows: a. More than \(10^{\circ}\) below average b. Normal to \(10^{\circ}\) below average c. Higher than normal to \(10^{\circ}\) above average d. More than \(10^{\circ}\) above average Using each state's abbreviation and the categories-(a), (b), (c), and (d) - the meteorologist records these data.

Evaluate the given expression. $$ P(5,3) $$