Chapter 14

If a game gives payoffs of \(\$ 10\) and \(\$ 100\) with probabilities 0.9 and \(0.1,\) respectively, then the expected value of this game is \(E=\) _____ \(\times 0.9+\) _____ \(\times 0.1\)

True or false? In counting combinations, order matters.

A binomial experiment is one in which there are exactly ______ outcomes. One outcome is called ______, and the other is called _____

The set of all possible outcomes of an experiment is called the _____ _____ A subset of the sample space is called an ____.

The Fundamental Counting Principle says that if one event can occur in \(m\) ways and a second event can occur in \(n\) ways, then the two events can occur in order in _____ x _____ ways. So if you have two choices for shoes and three choices for hats, then the number of different shoe-hat combinations you can wear is _____ x ____ = _____

True or false? In counting permutations, order matters.

The Fundamental Counting Principle also applies to three or more events in order. So if you have 2 choices for shoes, 5 choices for pants, 4 choices for shirts, and 3 choices for hats, then the number of different shoe-pants-shirt- hat outfits you can wear is ______x ______ x______x______ = ______

\(3-12\) . Find the expected value (or expectation) of the games described. Mike wins \(\$ 2\) if a coin toss shows heads and \(\$ 1\) if shows tails.

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

True or false? For a set of \(n\) distinct objects, the number of different combinations of these objects is more than the number of different permutations.

If the intersection of two events \(E\) and \(F\) is empty, then the events are called _____ ______ So in drawing a card from a deck, the event \(E\) of "getting a heart" and the event \(F\) of "getting a spade" are _____ _____.

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

\(3-12\) . Find the expected value (or expectation) of the games described. Jane wins \(\$ 10\) if a die roll shows a six, and she loses \(\$ 1\) otherwise.

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

True or false? If we have a set with five distinct objects, then the number of different ways of choosing two members of this set is the same as the number of ways of choosing three members.

If the occurrence of an event \(E\) does not affect the probability of the occurrence of another event \(F,\) then the events are called _____ So in tossing a coin, the event \(E\) of "getting heads on the first toss" and the event \(F\) of "getting heads on the _____ second toss" are _____.

Three-Letter Words How many three-letter "words" (strings of letters) can be formed by using the 26 letters of the alphabet if repetition of letters $$\begin{array}{lll}{\text { (a) is allowed? }} & {\text { (b) is not allowed? }}\end{array}$$

\(3-12\) . Find the expected value (or expectation) of the games described. The game consists of drawing a card from a deck. You win \(\$ 100\) if you draw the ace of spades or lose \(\$ 1\) if you draw any other 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. No successes

Evaluate the expression. $$ P(8,3) $$

An experiment consists of rolling a die. List the elements in the following sets. (a) The sample space (b) The event "getting an even number" (c) The event "getting a number greater than 4\("\)

Three-Letter Words How many three-letter "words" (strings of letters) can be formed by using the letters \(W X Y Z\) if repetition of letters $$\begin{array}{lll}{\text { (a) is allowed? }} & {\text { (b) is not allowed? }}\end{array}$$

\(3-12\) . Find the expected value (or expectation) of the games described. Tim wins \(\$ 3\) if a coin toss shows heads or \(\$ 2\) if it shows tails.

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

Evaluate the expression. $$ P(9,2) $$

An experiment consists of tossing a coin and drawing a card from a deck. (a) How many elements does the sample space have? (b) List the elements in the event "getting heads and an ace." (c) List the elements in the event "getting tails and a face card." (d) List the elements in the event "getting heads and a spade."

Horse Race 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 that there is no tie.)

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

Evaluate the expression. $$ P(11,4) $$

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.

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

\(3-12\) . Find the expected value (or expectation) of the games described. A coin is tossed twice. Albert wins \(\$ 2\) for each heads and must pay \(\$ 1\) for each tails.

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

Evaluate the expression. $$ P(10,5) $$

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.

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

\(3-12\) . Find the expected value (or expectation) of the games described. A die is rolled. Tom wins \(\$ 2\) if the die shows an even number, and he pays \(\$ 2\) otherwise.

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

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

\(9-10\) 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 five.

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

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

Evaluate the expression. $$ P(99,3) $$

\(9-10\) 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 \(3 .\)

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

\(3-12\) . Find the expected value (or expectation) of the games described. A bag contains two silver dollars and eight slugs. You pay 50 cents to reach into the bag and take a coin, which you get to keep.

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 number of distinguishable permutations of the given letters. $$ A A A B B C $$

\(11-12\) . 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.

\(3-12\) . Find the expected value (or expectation) of the games described. A bag contains eight white balls and two black balls. John picks two balls at random from the bag, and he wins \(\$ 5\) if he does not pick a black ball.