Chapter 4 Probability and Counting Rules

Chapter 4 Probability and Counting Rules Section 4-2 Sample Spaces and Probability

Section 4-2 Exercise #13 If two dice are rolled one time, find the probability of getting these results. • A sum of 6 • Doubles • A sum of 7 or 11 • A sum greater than 9 • A sum less than or equal to 4

Total of 36 outcomes • A sum of 6 • Doubles There are six ways to get doubles. They are (1,1), (2,2), (3,3), (4,4), (5,5), and (6,6).

Total of 36 outcomes • A sum of 7 or 11 There are six ways to get a sum of 7. They are (1,6), (2,5), (3,4), (4,3), (5,2), and (6,1). There are two ways to get a sum of 11. They are (5,6) and (6,5). • A sum of greater than 9 To get a sum greater than nine, one must roll a 10, 11, or 12. There are six ways to get a 10, 11, or 12. They are (4,6), (5,5), (6,4), (6,5), (5,6), and (6,6).

e. The patient has had 1 or 2 tests done.

Chapter 4 Probability and Counting Rules Section 4-3 Exercise #23

If one card is drawn from an ordinary deck of cards, find the probability of getting the following: • A king or a queen or a jack. • A club or a heart or a spade. • A king or a queen or a diamond. • An ace or a diamond or a heart. • A 9 or a 10 or a spade or a club.

P (king or queen or jack) If one card is drawn from an ordinary deck of cards, find the probability of getting the following: • A king or a queen or a jack. There are 4 kings, 4 queens, and 4 jacks, hence:

If one card is drawn from an ordinary deck of cards, find the probability of getting the following: • A club or a heart or a spade. There are 13 clubs, 13 hearts, and 13 spades, hence: P(club or heart or spade)

P(king or queen or diamond) P(king) + P(queen) + P(diamond) – P(king or queen of diamonds) If one card is drawn from an ordinary deck of cards, find the probability of getting the following: • A king or a queen or a diamond. There are 4 kings, 4 queens, and 13 diamonds but the king and queen of diamonds were counted twice, hence:

P(ace or diamond or heart) If one card is drawn from an ordinary deck of cards, find the probability of getting the following: • An ace or a diamond or a heart. There are 4 aces, 13 diamonds and 13 hearts. There is one ace of diamonds and one ace of hearts, hence:

P ( 9 or 10 or spade or club) If one card is drawn from an ordinary deck of cards, find the probability of getting the following: • A 9 or a 10 or a spade or a club. There are 4 nines, 4 tens, 13 spades, and 13 clubs. There is one nine of spades, one ten of spades, one nine of clubs, and one ten of clubs, hence:

P ( 9 or 10 or spade or club) If one card is drawn from an ordinary deck of cards, find the probability of getting the following: • A 9 or a 10 or a spade or a club.

Chapter 4 Probability and Counting Rules Section 4-4 The Multiplication Rules and Conditional Probability

At a local university 54.3% of incoming first-year students have computers. If three students are selected at random, find the following probabilities. • None have computers • At least one has a computer • All have computers

At a local university 54.3% of incoming first-year students have computers. If three students are selected at random, find the following probabilities. • None have computers

At a local university 54.3% of incoming first-year students have computers. If three students are selected at random, find the following probabilities. b. At least one has a computer

At a local university 54.3% of incoming first-year students have computers. If three students are selected at random, find the following probabilities. c. All have computers

0.1 D (0.8)(0.1) = 0.08 0.8 0.9 ND 0.18 D (0.2)(0.18) = 0.036 0.2 0.82 ND

0.1 D (0.8)(0.1) = 0.08 0.8 0.9 ND 0.18 D (0.2)(0.18) = 0.036 0.2 0.82 ND Finally, use the addition rule, since the item is chosen at random from model I or model II.

In Rolling Acres Housing Plan, 42% of the houses have a deck and a garage; 60% have a deck. Find the probability that a home has a garage, given that it has a deck.

Consider this table concerning utility patents granted for a specific year. • Select one patent at random. • What is the probability that it is a • foreign patent, given that it was • issued to a corporation? • What is the probability that it was issued to an individual, given that it was a U.S. patent?

P(foreign patent | corporation) • What is the probability that it is a • foreign patent, given that it was • issued to a corporation?

P(foreign patent | corporation)

P (individual | U.S.) What is the probability that it was issued to an individual, given that it was a U.S. patent?

P(individual | U.S.)

Chapter 4 Probability and Counting Rules Section 4-5 Counting Rules

How many different 3 - digit identification tags can be made if the digits can be used more than once? If the first digit must be a 5 and repetitions are not permitted? If digits can be used more than once: Since there are three spaces to fill and 10 choices for each space, the solution is:

How many different 3 - digit identification tags can be made if the digits can be used more than once? If the first digit must be a 5 and repetitions are not permitted? If the first digit must be a 5 and repetitions are not permitted: There is only one way to assign the first digit, 9 ways to assign the second, and 8 ways to assign the third:

Since order is important, the solution is: How many different ID cards can be made if there are 6 digits on a card and no digit can be used more than once?

Since order is not important, the solution is: How many ways can a committee of 4 people be selected from a group of 10 people?

How many ways can a foursome of 2 men and 2 women be selected from 10 men and 12 women in a golf club?

Chapter 4 Probability and Counting Rules Section 4-6 Probability and Counting Rules

In a company there are 7 executives: 4 women and 3 men. Three are selected to attend a management seminar. Find these probabilities. • All 3 selected will be women. • All 3 selected will be men. • c. 2 men and 1 woman will be selected. • d. 1 man and 2 women will be selected.

In a company there are 7 executives: 4 women and 3 men. Three are selected to attend a management seminar. Find these probabilities. • All 3 selected will be women.

In a company there are 7 executives: 4 women and 3 men. Three are selected to attend a management seminar. Find these probabilities. • All 3 selected will be men.

In a company there are 7 executives: 4 women and 3 men. Three are selected to attend a management seminar. Find these probabilities. c. 2 men and 1 woman will be selected.

In a company there are 7 executives: 4 women and 3 men. Three are selected to attend a management seminar. Find these probabilities. d. 1 man and 2 women will be selected.

Chapter 4 Probability and Counting Rules