Chapter 13

Five independent trials of a binomial experiment with probability of success \(p=0.7\) and probability of failure \(q=0.3\) are performed. Find the probability of each event. Exactly two successes

Find the expected value (or expectation) of the games described.? $$\begin{array}{l}{\text { Mike wins } \$ 2 \text { if a coin toss shows heads and } \$ 1 \text { if it shows }} \\ {\text { tails. }}\end{array}$$

A vendor sells ice cream from a cart on the boardwalk. He offers vanilla, chocolate, strawberry, and pistachio ice cream, served on either a waffle, sugar, or plain cone. How many different single-scoop ice-cream cones can you buy from this vendor?

1–6 Evaluate the expression. $$P(8,3)$$

An experiment consists of tossing a coin twice. (a) Find the sample space. (b) Find the probability of getting heads exactly two times. (c) Find the probability of getting heads at least one time. (d) Find the probability of getting heads exactly one time.

Five independent trials of a binomial experiment with probability of success \(p=0.7\) and probability of failure \(q=0.3\) are performed. Find the probability of each event. Exactly three successes

Find the expected value (or expectation) of the games described.? $$ \begin{array}{l}{\text { Jane wins } \$ 10 \text { if a die roll shows a six, and she loses } \$ 1} \\ {\text { otherwise. }}\end{array} $$

How many three-letter “words” (strings of letters) can be formed using the 26 letters of the alphabet if repetition of letters (a) is allowed? (b) is not allowed?

1–6 Evaluate the expression. $$P(9,2)$$

An experiment consists of tossing a coin and rolling a die. (a) Find the sample space. (b) Find the probability of getting heads and an even number. (c) Find the probability of getting heads and a number greater than 4. (d) Find the probability of getting tails and an odd number.

Five independent trials of a binomial experiment with probability of success \(p=0.7\) and probability of failure \(q=0.3\) are performed. Find the probability of each event. No successes

Find the expected value (or expectation) of the games described.? $$ \begin{array}{l}{\text { The game consists of drawing a card from a deck. You win }} \\ {\$ 100 \text { if you draw the ace of spades or lose } \$ 1 \text { if you draw }} \\ {\text { any other card. }}\end{array} $$

How many three-letter “words” (strings of letters) can be formed using the letters WXYZ if repetition of letters (a) is allowed? (b) is not allowed?

1–6 Evaluate the expression. $$P(11,4)$$

A die is rolled. Find the probability of the given event. (a) The number showing is a six. (b) The number showing is an even number. (c) The number showing is greater than 5.

Five independent trials of a binomial experiment with probability of success \(p=0.7\) and probability of failure \(q=0.3\) are performed. Find the probability of each event. All successes

1–6 Evaluate the expression. $$P(10,5)$$

Eight horses are entered in a race. (a) How many different orders are possible for completing the race? (b) In how many different ways can first, second, and third places be decided? (Assume there is no tie.)

A die is rolled. Find the probability of the given event. (a) The number showing is a two or a three. (b) The number showing is an odd number. (c) The number showing is a number divisible by 3.

Five independent trials of a binomial experiment with probability of success \(p=0.7\) and probability of failure \(q=0.3\) are performed. Find the probability of each event. Exactly one success

Find the expected value (or expectation) of the games described.? $$ \begin{array}{l}{\text { Carol wins } \$ 3 \text { if a die roll shows a six, and she wins } \$ 0.50} \\ {\text { otherwise. }}\end{array} $$

1–6 Evaluate the expression. $$P(100,1)$$

A multiple-choice test has five questions with four choices for each question. In how many different ways can the test be completed?

A card is drawn randomly from a standard 52-card deck. Find the probability of the given event. (a) The card drawn is a king. (b) The card drawn is a face card. (c) The card drawn is not a face card.

Five independent trials of a binomial experiment with probability of success \(p=0.7\) and probability of failure \(q=0.3\) are performed. Find the probability of each event. Exactly one failure

Find the expected value (or expectation) of the games described.? $$ \begin{array}{l}{\text { A coin is tossed twice. Albert wins } \$ 2 \text { for each heads and }} \\ {\text { must pay } \$ 1 \text { for each tails. }}\end{array} $$

1–6 Evaluate the expression. $$P(99,3)$$

Telephone numbers consist of seven digits; the first digit cannot be 0 or 1. How many telephone numbers are possible?

A card is drawn randomly from a standard 52-card deck. Find the probability of the given event. (a) The card drawn is a heart. (b) The card drawn is either a heart or a spade. (c) The card drawn is a heart, a diamond, or a spade.

Five independent trials of a binomial experiment with probability of success \(p=0.7\) and probability of failure \(q=0.3\) are performed. Find the probability of each event. At least four successes

Find the expected value (or expectation) of the games described.? $$ \begin{array}{l}{\text { A die is rolled. Tom wins } \$ 2 \text { if the die shows an even num- }} \\ {\text { ber and he pays } \$ 2 \text { otherwise. }}\end{array} $$

7–12 Find the number of distinguishable permutations of the given letters. $$A A A B B C$$

In how many different ways can a race with five runners be completed? (Assume there is no tie.)

A ball is drawn randomly from a jar that contains five red balls, two white balls, and one yellow ball. Find the probability of the given event. (a) A red ball is drawn. (b) The ball drawn is not yellow. (c) A black ball is drawn.

Five independent trials of a binomial experiment with probability of success \(p=0.7\) and probability of failure \(q=0.3\) are performed. Find the probability of each event. At least three successes

Find the expected value (or expectation) of the games described.? $$ \begin{array}{l}{\text { A card is drawn from a deck. You win } \$ 104 \text { if the card is an }} \\ {\text { ace, } \$ 26 \text { if it is a face card, and } \$ 13 \text { if it is the } 8 \text { of clubs. }}\end{array} $$

7–12 Find the number of distinguishable permutations of the given letters. $$A A A B B B C C C$$

In how many ways can five people be seated in a row of five seats?

A ball is drawn randomly from a jar that contains five red balls, two white balls, and one yellow ball. Find the probability of the given event. (a) Neither a white nor yellow ball is drawn. (b) A red, white, or yellow ball is drawn. (c) The ball drawn is not white.

Five independent trials of a binomial experiment with probability of success \(p=0.7\) and probability of failure \(q=0.3\) are performed. Find the probability of each event. At most one failure

Find the expected value (or expectation) of the games described.? $$ \begin{array}{l}{\text { A bag contains two silver dollars and eight slugs. You pay }} \\ {50 \text { cents to reach into the bag and take a coin, which you get }} \\ {\text { to keep. }}\end{array} $$

7–12 Find the number of distinguishable permutations of the given letters. $$A A B C D$$

A drawer contains an unorganized collection of 18 socks— three pairs are red, two pairs are white, and four pairs are black. (a) If one sock is drawn at random from the drawer, what is the probability that it is red? (b) Once a sock is drawn and discovered to be red, what is the probability of drawing another red sock to make a matching pair?

Five independent trials of a binomial experiment with probability of success \(p=0.7\) and probability of failure \(q=0.3\) are performed. Find the probability of each event. At most two failures

Find the expected value (or expectation) of the games described.? $$ \begin{array}{l}{\text { A bag contains eight white balls and two black balls. John }} \\ {\text { picks two balls at random from the bag, and he wins } \$ 5 \text { if he }} \\ {\text { does not pick a black ball. }}\end{array} $$

7–12 Find the number of distinguishable permutations of the given letters. $$A B C D D D E E$$

In how many ways can five different mathematics books be placed next to each other on a shelf?

Five independent trials of a binomial experiment with probability of success \(p=0.7\) and probability of failure \(q=0.3\) are performed. Find the probability of each event. At least two successes

Roulette In the game of roulette as played in Las Vegas, the wheel has 38 slots: Two slots are numbered 0 and 00 , and the rest are numbered 1 to \(36 .\) A \(\$ 1\) bet on any number other than 0 or 00 wins \(\$ 36\) (\$35 plus the \(\$ 1\) bet). Find the expected value of this game.

7–12 Find the number of distinguishable permutations of the given letters. $$X X Y Y Y Z Z Z$$