1. 11-1 Permutations and Combinations

2. Warm Up
Evaluate. 4
210 10 70

4. Example 1A: Using the Fundamental Counting Principle
To make a yogurt parfait, you choose one flavor of yogurt, one fruit topping, and one nut topping. How many parfait choices are there?
Yogurt Parfait
(choose 1 of each)
Flavor
Plain
Vanilla
Fruit
Peaches
Strawberries
Bananas
Raspberries
Blueberries
Nuts
Almonds
Peanuts
Walnuts

5. Example 1A Continued
number
of flavors
number
of fruits
number
of nuts
number
of choices
times
times
equals
2   = 30
There are 30 parfait choices.

6. Example 1B: Using the Fundamental Counting Principle
A password for a site consists of 4 digits followed by 2 letters. The letters A and Z are not used, and each digit or letter many be used more than once. How many unique passwords are possible?
digit digit digit digit letter letter
10  10  10    = 5,760,000
There are 5,760,000 possible passwords.

7. number of starting points
Check It Out! Example 1a
A “make-your-own-adventure” story lets you choose 6 starting points, gives 4 plot choices, and then has 5 possible endings. How many adventures are there?
number of starting points
number
of possible endings
number
of plot choices
number
of adventures   =
6   =
There are 120 adventures.

8. A permutation is a selection of a group of objects in which order is important.
A combination is a grouping of items in which order does not matter. There are generally fewer ways to select items when order does not matter.
6 permutations  {ABC, ACB, BAC, BCA, CAB, CBA}
1 combination  {ABC}

12. When deciding whether to use permutations or combinations, first decide whether order is important. Use a permutation if order matters and a combination if order does not matter.

13. How many ways can a student government select a president, vice president, secretary, and treasurer from a group of 6 people?
This is the equivalent of selecting and arranging 4 items from 6.
Substitute 6 for n and 4 for r in
Divide out common factors.
= 6 • 5 • 4 • 3 = 360
There are 360 ways to select the 4 people.

14. There are 12 different-colored cubes in a bag
There are 12 different-colored cubes in a bag. How many ways can Randall draw a set of 4 cubes from the bag?
Step 1 Determine whether the problem represents a permutation of combination.
The order does not matter. The cubes may be drawn in any order. It is a combination.

15. Step 2 Use the formula for combinations.
Example 3 Continued
Step 2 Use the formula for combinations.
n = 12 and r = 4
Divide out
common
factors. 5
= 495
There are 495 ways to draw 4 cubes from 12.

16. Check It Out! Example 2a
Awards are given out at a costume party. How many ways can “most creative,” “silliest,” and “best” costume be awarded to 8 contestants if no one gets more than one award?
= 8 • 7 • 6
= 336
There are 336 ways to arrange the awards.

17. Check It Out! Example 2b
How many ways can a 2-digit number be formed by using only the digits 5–9 and by each digit being used only once?
= 5 • 4
= 20
There are 20 ways for the numbers to be formed.

18. The swimmers can be selected in 28 ways.
Check It Out! Example 3
The swim team has 8 swimmers. Two swimmers will be selected to swim in the first heat. How many ways can the swimmers be selected?
n = 8 and r = 2
Divide out
common
factors. 4
= 28
The swimmers can be selected in 28 ways.

19. Lesson Quiz
1. Six different books will be displayed in the library window. How many different arrangements are there?
2. The code for a lock consists of 5 digits. The last number cannot be 0 or 1. How many different codes are possible?
720
80,000
3. The three best essays in a contest will receive gold, silver, and bronze stars. There are 10 essays. In how many ways can the prizes be awarded?
4. In a talent show, the top 3 performers of 15 will advance to the next round. In how many ways can this be done?
720
455

20. Experimental Probability Vs. Theoretical Probability
Lesson 11.2

21. What do you know about probability?
Probability is a number from 0 to 1 that tells you how likely something is to happen.
Probability can have two approaches -experimental probability
-theoretical probability

22. Experimental vs.Theoretical
Experimental probability:
P(event) = number of times event occurs
total number of trials
Theoretical probability:
P(E) = number of favorable outcomes total number of possible outcomes

23. How can you tell which is experimental and which is theoretical probability?
You tossed a coin 10 times and recorded a head 3 times, a tail 7 times
P(head)= 3/10
P(tail) = 7/10
Theoretical:
Toss a coin and getting a head or a tail is 1/2.
P(head) = 1/2
P(tail) = 1/2

24. Experimental probability
Experimental probability is found by repeating an experiment and observing the outcomes.
P(head)= 3/10
A head shows up 3 times out of 10 trials,
P(tail) = 7/10
A tail shows up 7 times out of 10 trials

25. Theoretical probability
P(head) = 1/2
P(tail) = 1/2
Since there are only two outcomes, you have 50/50 chance to get a head or a tail.
HEADS
TAILS

26. Compare experimental and theoretical probability
Both probabilities are ratios that compare the number of favorable outcomes to the total number of possible outcomes
P(head)= 3/10
P(tail) = 7/10
P(head) = 1/2
P(tail) = 1/2

27. Identifying the Type of Probability
A bag contains three red marbles and three blue marbles.
P(red) = 3/6 =1/2
Theoretical
(The result is based on the possible outcomes)

28. Identifying the Type of Probability
You draw a marble out of the bag, record the color, and replace the marble. After 6 draws, you record 2 red marbles
P(red)= 2/6 = 1/3
Experimental
(The result is found by repeating an experiment.)

29. How come I never get a theoretical value in both experiments? Tom asked.
If you repeat the experiment many times, the results will getting closer to the theoretical value.
Law of the Large Numbers

30. Law of the Large Numbers 101
The Law of Large Numbers was first published in 1713 by Jocob Bernoulli.
It is a fundamental concept for probability and statistic.
This Law states that as the number of trials increase, the experimental probability will get closer and closer to the theoretical probability.

31. Contrast experimental and theoretical probability
Experimental probability is the result of an experiment.
Theoretical probability is what is expected to happen.

32. Contrast Experimental and theoretical probability
Three students tossed a coin 50 times individually.
Lisa had a head 20 times. ( 20/50 = 0.4)
Tom had a head 26 times. ( 26/50 = 0.52)
Al had a head 28 times. (28/50 = 0.56)
Please compare their results with the theoretical probability.
It should be 25 heads. (25/50 = 0.5)

33. Contrast Experimental and theoretical probability

34. Probability is the measure of how likely an event is to occur
Probability is the measure of how likely an event is to occur. Each possible result of a probability experiment or situation is an outcome. The sample space is the set of all possible outcomes. An event is an outcome or set of outcomes.

35. Probabilities are written as fractions or decimals from 0 to 1, or as percents from 0% to 100%.

36. Equally likely outcomes have the same chance of occurring
Equally likely outcomes have the same chance of occurring. When you toss a fair coin, heads and tails are equally likely outcomes. Favorable outcomes are outcomes in a specified event. For equally likely outcomes, the theoretical probability of an event is the ratio of the number of favorable outcomes to the total number of outcomes.

37. Example 1A: Finding Theoretical Probability
Each letter of the word PROBABLE is written on a separate card. The cards are placed face down and mixed up. What is the probability that a randomly selected card has a consonant?
There are 8 possible outcomes and 5 favorable outcomes.

38. Example 1B: Finding Theoretical Probability
Two number cubes are rolled. What is the probability that the difference between the two numbers is 4?
There are 36 possible outcomes.
4 outcomes with a difference of 4: (1, 5), (2, 6), (5, 1), and (6, 2)

39. Check It Out! Example 1a
A red number cube and a blue number cube are rolled. If all numbers are equally likely, what is the probability of the event?
The sum is 6.

40. Check It Out! Example 1b
A red number cube and a blue number cube are rolled. If all numbers are equally likely, what is the probability of the event?
The difference is 6.

41. Check It Out! Example 1c
A red number cube and a blue number cube are rolled. If all numbers are equally likely, what is the probability of the event?
The red cube is greater.

42. The sum of all probabilities in the sample space is 1
The sum of all probabilities in the sample space is 1. The complement of an event E is the set of all outcomes in the sample space that are not in E.

43. Language Number French 6 Spanish 12 Japanese 3 Example 2: Application
There are 25 students in study hall. The table shows the number of students who are studying a foreign language. What is the probability that a randomly selected student is not studying a foreign language?
Language
Number
French 6
Spanish 12
Japanese 3

44. Example 2 Continued
P(not foreign) = 1 – P(foreign)
Use the complement.
There are 21 students studying a foreign language.
, or 16%
There is a 16% chance that the selected student is not studying a foreign language.

45. Check It Out! Example 2
Two integers from 1 to 10 are randomly selected. The same number may be chosen twice. What is the probability that both numbers are less than 9?

46. Example 3: Finding Probability with Permutations or Combinations
Each student receives a 5-digit locker combination. What is the probability of receiving a combination with all odd digits?
Step 1 Determine whether the code is a permutation or a combination.
Order is important, so it is a permutation.

47. Example 3 Continued
Step 2 Find the number of outcomes in the sample space.
number number number number number
10     = 100,000
There are 100,000 outcomes.

48. Example 3 Continued
Step 3 Find the number of favorable outcomes.
odd odd odd odd odd
5  5  5  5  = 3125
There are 3125 favorable outcomes.

49. Example 3 Continued
Step 4 Find the probability.
The probability that a combination would have only odd digits is

50. Check It Out! Example 3
A DJ randomly selects 2 of 8 ads to play before her show. Two of the ads are by a local retailer. What is the probability that she will play both of the retailer’s ads before her show?

51. Geometric probability is a form of theoretical probability determined by a ratio of lengths, areas, or volumes.

52. Example 4: Finding Geometric Probability
A figure is created placing a rectangle inside a triangle inside a square as shown. If a point inside the figure is chosen at random, what is the probability that the point is inside the shaded region?

53. Example 4 Continued
Find the ratio of the area of the shaded region to the area of the entire square. The area of a square is s2, the area of a triangle is , and the area of a rectangle is lw.
First, find the area of the entire square.
At = (9)2 = 81
Total area of the square.

54. Example 4 Continued
Next, find the area of the triangle.
Area of the triangle.
Next, find the area of the rectangle.
Arectangle = (3)(4) = 12
Area of the rectangle.
Subtract to find the shaded area.
As = 40.5 – 12 = 28.5
Area of the shaded region.
Ratio of the shaded region to total area.

55. Check It Out! Example 4
Find the probability that a point chosen at random inside the large triangle is in the small triangle.

56. You can estimate the probability of an event by using data, or by experiment. For example, if a doctor states that an operation “has an 80% probability of success,” 80% is an estimate of probability based on similar case histories.
Each repetition of an experiment is a trial. The sample space of an experiment is the set of all possible outcomes. The experimental probability of an event is the ratio of the number of times that the event occurs, the frequency, to the number of trials.

57. Experimental probability is often used to estimate
theoretical probability and to make predictions.

58. Example 5A: Finding Experimental Probability
The table shows the results of a spinner experiment. Find the experimental probability.
Number
Occurrences 1 6 2 11 3 19 4 14
spinning a 4
The outcome of 4 occurred 14 times out of 50 trials.

59. Example 5B: Finding Experimental Probability
The table shows the results of a spinner experiment. Find the experimental probability.
spinning a number greater than 2
Number
Occurrences 1 6 2 11 3 19 4 14
The numbers 3 and 4 are greater than 2.
3 occurred 19 times and 4 occurred 14 times.

60. Check It Out! Example 5a
The table shows the results of choosing one card from a deck of cards, recording the suit, and then replacing the card.
Find the experimental probability of choosing a diamond.

61. Check It Out! Example 5b
The table shows the results of choosing one card from a deck of cards, recording the suit, and then replacing the card.
Find the experimental probability of choosing a card that is not a club.

65. 11.3 Probability of Multiple Events
Learning goal
find the probability of the event A and B
find the probability of the event A or B

66. Vocabulary
dependent events : when the outcome of one event affects the outcome of a second event
independent events : when the outcome of one event does not affect the outcome of the second event

67. Ex 1 Classify as dependent or independent
Spin a spinner. Then, select a marble from a bag that contains marbles of different colors.
Select a marble from a bag that contains marbles of two colors. Put the marble aside, and select a second marble from the bag.
What if we put the marble back???

68. If A and B are independent events, then P(A and B) = P(A)●P(B)
Probability of A and B
If A and B are independent events, then P(A and B) = P(A)●P(B)

69. Ex 2
A box contains 20 red marbles and 30 blue marbles. A second box contains 10 white marbles and 47 black marbles. If you choose one marble from each box without looking, what is the probability that you get a blue marble and a black marble?

70. Vocabulary
mutually exclusive events : when two events cannot happen at the same time.
If A and B are mutually exclusive events, then P(A and B) = 0.

71. Ex 3 Are the events mutually exclusive?
Rolling an even number and rolling a number greater than 5 on a number cube?
Rolling a prime number and a multiple of 6 on a number cube?
Rolling an even number and rolling a number less than 2 on a number cube.

72. Probability of A or B
If A & B are mutually exclusive events, then P(A or B) = P(A) + P(B)
If A & B are NOT mutually exclusive events, then P(A or B) = P(A) + P(B) – P(A and B)

73. Ex 4
At a restaurant, customers get to choose one of four vegetables with any main course. About 33% of the customers choose green beans, and about 28% choose spinach. What is the probability that a customer will choose beans or spinach?

74. Ex 5
A spinner has twenty equal size sections numbered from 1 to 20. If you spin the spinner, what is the probability that the number you spin will be a multiple of 2 or a multiple of 3?

75. Ex 6 What is the probability of selecting a square or a red token?
What is the probability of selecting a token that is green or a square?

76. Ex 7 Find the probability of selecting a boy or a blond person from 12 girls, 5 of whom have blond hair, and 15 boys, 6 of whom have blond hair.

77. Ex 8
A bank contains 4 nickels, 4 dimes and 7 quarters. Three coins are removed in sequence without replacement. What is the probability of selecting a nickel, a dime and a quarter in that order?

78. Ex 9 Given a standard deck of cards, find
P(5 or Jack)
P(king or spade)
P(both red or both queens) if 2 cards are drawn without replacement

79. Ex 10
There are 5 male and 5 female students. A committee of 4 members is to be selected at random. Find P(all female).