Chapter 16

Explain why the expansion of \((x+y)^{n}=\sum_{i=0}^{n} C_{i} x^{n-i} y^{i}\) can also be written as \((x+y)^{n}=\sum_{i=0}^{n} C_{n-i} x^{n-i} y^{i}\)

There are 20 students in a club, 12 boys and 8 girls. If five members of the club are chosen at random to represent the club at a competition, what is the probability that in the group chosen there are exactly 2 boys? Explain why this is not a Bernoulli experiment.

Emma said that if the probability of success in at least \(r\) out of \(n\) trials is \(z,\) then the probability of success in at most \(r\) out of \(n\) trials is \(1-z .\) Do you agree with Emma? Explain why or why not.

A seed company tests \(2,000\) tomato seeds and obtains \(1,954\) plants. The seed company advertises that their seed is 97\(\%\) productive. a. Is the seed company's claim justified? b. Is the seed company's claim based on theoretical or empirical probability?

Show that \(_{n} C_{r}=\frac{n !}{(n-r) ! \times r !}\)

How is choosing a boy and a girl from 12 boys and 12 girls to represent a club different from choosing two girls from 12 girls to be president and treasurer of the club?

Hunter said that the number of combinations of \(n\) things taken \(r\) at a time is equal to the number of permutations of \(n\) things taken \(n\) at a time when \(r\) are identical; that is, \(C_{r}=\frac{n !}{r !}\) . Do you agree with Hunter? Explain why or why not.

Compare the number of ordered pairs of cards that can be drawn from a deck of 52 cards without replacement to the number of ordered pairs of cards that can be drawn from a deck of 52 cards with replacement.

In \(3-10,\) write the expansion of each binomial. $$ (x+y)^{6} $$

In \(3-6,\) find exact probabilities showing all required computation. A fair coin is tossed five times. What is the probability that the coin lands heads: $$ \begin{array}{lll}{\text { a. exactly once? }} \quad {\text { b. exactly twice? }} \quad {\text { c. exactly three times? }} \\ {\text { d. exactly four times? }} \quad {\text { e. exactly five times? }} \quad {\text { f. zero times? }} \\ {\text { g. Which is the most likely event(s) when tossing a coin five times? }}\end{array} $$

In \(3-5,\) find exact probabilities showing all required computation. Five fair coins are tossed. Find the probability of the coins showing: \(\begin{array}{ll}{\text { a. at least four heads }} & {\text { b. at least three heads }} \\ {\text { c. at least two heads }} & {\text { d. at least one head }}\end{array}\)

What is the probability of getting a 2 on a single throw of a fair die?

In \(3-22,\) evaluate each expression. $$ 5 ! $$

For the given values of \(r\) and \(n,\) find the number of ordered selections of \(r\) objects from a collection of \(n\) objects without replacement. \(r=4, n=4\)

In \(3-10,\) write the expansion of each binomial. $$ (x+y)^{7} $$

In \(3-6,\) find exact probabilities showing all required computation. A fair die is tossed five times. What is the probability of tossing a \(6 :\) $$ \begin{array}{llll}{\text { a. exactly once? }} & {\text { b. exactly twice? }} & {\text { c. exactly three times? }} \\ {\text { d. exactly four times? }} & {\text { e. exactly five times? }} & {\text { f. zero times? }}\end{array} $$ g. Which is the most likely event(s) when tossing a die five times?

In \(3-5,\) find exact probabilities showing all required computation. Three fair dice are tossed. Find the probability of the dice showing: \(\begin{array}{ll}{\text { a. at most one } 6} & {\text { b. at most two } 6 \text { s }} \\ {\text { c. at least two } 6^{\prime} \mathrm{s}} & {\text { d. at least one } 6}\end{array}\)

What is the probability of getting a number greater than 2 on a single throw of a fair die?

In \(3-22,\) evaluate each expression. $$ 12 ! $$

For the given values of \(r\) and \(n,\) find the number of ordered selections of \(r\) objects from a collection of \(n\) objects without replacement. \(r=4, n=5\)

In \(3-10,\) write the expansion of each binomial. $$ (1+y)^{5} $$

In \(3-6,\) find exact probabilities showing all required computation. A card is drawn and replaced four times from a standard deck of 52 cards. What is the probability of drawing a king: a. exactly once? \(\quad\) b. exactly twice? c. exactly three times? \(\quad\) d. exactly four times? \(\quad\) e. zero times? f. Which is the most likely event(s) when drawing a card four times with replacement?

A spinner is divided into five equal sections numbered 1 through \(5 .\) The arrow is equally likely to land on any section. Find the probability of: a. an odd number on any one spin b. at least three odd numbers on four spins c. at least two odd numbers on four spins d. at least one odd number on four spins

What is the probability of getting a sum of 8 when a pair of dice are thrown?

In \(3-22,\) evaluate each expression. $$ 8 ! \div 3 ! $$

For the given values of \(r\) and \(n,\) find the number of ordered selections of \(r\) objects from a collection of \(n\) objects without replacement. \(r=3, n=8\)

In \(3-10,\) write the expansion of each binomial. $$ (x+2)^{5} $$

In \(3-6,\) find exact probabilities showing all required computation. A multiple-choice test of 10 questions has 4 choices for each question. Only one choice is correct. A student who did not study for the test guesses at each answer. a. What is the probability that that student will have exactly 5 correct answers? b. What is the probability that that student will have only 1 correct answer?

In \(6-9,\) write an expression using sigma notation that can be used to find each probability. At least 10 heads when 15 coins are tossed

The buyer of a lottery ticket chooses four numbers from the numbers 1 to \(32 .\) Repetition is not allowed. a. How many combinations of four numbers are possible? b. What is the probability of choosing all four of the winning numbers?

In \(3-22,\) evaluate each expression. $$ 9 ! \div 8 ! $$

For the given values of \(r\) and \(n,\) find the number of ordered selections of \(r\) objects from a collection of \(n\) objects without replacement. \(r=2, n=12\)

In \(3-10,\) write the expansion of each binomial. $$ (a+3)^{4} $$

In \(7-14,\) answers can be rounded to four decimal places. The probability that our team will win a basketball game is \(\frac{2}{3} .\) What is the probability that they will win exactly 5 of the next 7 games?

In \(6-9,\) write an expression using sigma notation that can be used to find each probability. At most 7 tails when 10 coins are tossed

A standard deck of cards contains 52 cards divided into 4 suits. There are two red suits: hearts and diamonds, and two black suits: clubs and spades. Each suit contains 13 cards; ace, king, queen, jack, and cards numbered 2 through \(10 .\) A card is drawn from a standard deck without replacement. What is the probability that the card is a king?

In \(3-22,\) evaluate each expression. $$ \frac{10 !}{3 !} $$

For the given values of \(r\) and \(n,\) find the number of ordered selections of \(r\) objects from a collection of \(n\) objects with replacement. \(r=4, n=4\)

In \(3-10,\) write the expansion of each binomial. $$ (2+a)^{4} $$

In \(6-9,\) write an expression using sigma notation that can be used to find each probability. At least 5 wins in the next 20 games when \(P(\text { win })=\frac{2}{3}\)

Two cards are drawn from a standard deck of 52 cards without replacement. What is the probability that both cards are kings?

In \(3-22,\) evaluate each expression. $$ _{6} P_{6} $$

For the given values of \(r\) and \(n,\) find the number of ordered selections of \(r\) objects from a collection of \(n\) objects with replacement. \(r=4, n=5\)

In \(3-10,\) write the expansion of each binomial. $$ (2 b-1)^{3} $$

Three cards are drawn from a standard deck of 52 cards without replacement. What is the probability that all three cards are kings?

In \(3-22,\) evaluate each expression. $$ _{8} P_{4} $$

For the given values of \(r\) and \(n,\) find the number of ordered selections of \(r\) objects from a collection of \(n\) objects with replacement. \(r=2, n=8\)

In \(3-10,\) write the expansion of each binomial. $$ (1-i)^{5} ; i=\sqrt{-1} $$

In \(7-14,\) answers can be rounded to four decimal places. A fast-food restaurant gives coupons for 10\(\%\) off of the next purchase with 1 out of every 5 purchases. a. What is the probability that Zoe will receive a coupon with her next purchase? b. What is the probability that Zoe will receive just one coupon with her next three purchases?

In \(10-13,\) the mean and standard deviation of a normal distribution are given. Find each probability to the nearest hundredth. mean \(=80,\) standard deviation \(=10, P(50 \leq x \leq 95)\)