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19.2 Permutations and Probability....2016/08/19  · 19.2 Permutations and Probability. 19.3 Combinations and Probability. Use permutations and combinations to compute probabilities

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Page 1: 19.2 Permutations and Probability....2016/08/19  · 19.2 Permutations and Probability. 19.3 Combinations and Probability. Use permutations and combinations to compute probabilities
Page 2: 19.2 Permutations and Probability....2016/08/19  · 19.2 Permutations and Probability. 19.3 Combinations and Probability. Use permutations and combinations to compute probabilities

19.2 Permutations and Probability. 19.3 Combinations and Probability.

Use permutations and combinations to compute probabilities of compound events and solve problems.

When are permutations useful in calculating probability?

What is the difference between a permutacion and a combination?

Page 3: 19.2 Permutations and Probability....2016/08/19  · 19.2 Permutations and Probability. 19.3 Combinations and Probability. Use permutations and combinations to compute probabilities

What is a permutation?

A permutation is a selection of objects from a group in which order is important. For example, there are 6 permutations of the letters A, B, and C.

Fundamental Counting Principle: If you have A ways of doing event 1,B ways of doing event 2, and C ways of event 3, then you can find the total number of outcomes by multiplying:

𝑨 × 𝑩 × 𝑪

This principle is difficult to explain in words. To find the total number of outcomes for the scenario, multiply the total outcomes for each individual event.

For example: 3 choices of sandwiches3 choices of sides2 choices of drinks

3 × 3 × 2 = 18 𝑡𝑜𝑡𝑎𝑙 𝑜𝑢𝑡𝑐𝑜𝑚𝑒𝑠

Page 4: 19.2 Permutations and Probability....2016/08/19  · 19.2 Permutations and Probability. 19.3 Combinations and Probability. Use permutations and combinations to compute probabilities

There are 7 members in a club. Each year the club elects a president, a vice president, and a treasurer.

76

5

7 × 6 × 5 × 4 × 3 × 2 × 1 = 5040

7

65

7 × 6 × 5 = 210

Page 5: 19.2 Permutations and Probability....2016/08/19  · 19.2 Permutations and Probability. 19.3 Combinations and Probability. Use permutations and combinations to compute probabilities

4 4

3

4 × 3 × 2 × 1 = 24

5040

24𝟓𝟎𝟒𝟎

𝟐𝟒= 210

Page 6: 19.2 Permutations and Probability....2016/08/19  · 19.2 Permutations and Probability. 19.3 Combinations and Probability. Use permutations and combinations to compute probabilities

Finding a Probability Using Permutations

What is factorial?

4! Is usually pronounced “4 factorial”, but some people even say “4 shriek” or “4 bang”

That is, n! is the product of n and all the positive integers less than n.Note that 0! is defined to be 1.

Page 7: 19.2 Permutations and Probability....2016/08/19  · 19.2 Permutations and Probability. 19.3 Combinations and Probability. Use permutations and combinations to compute probabilities

Another way to write it …

Permutations

GC have a built-in function that calculates permutations.

For example, to find 10P4 , first enter 10. Then press MATH and use the arrow keys to

choose the PRB menu. Select 2:nPr and press ENTER. Now enter 4 and then press ENTER to see that 𝑃(10,4) = 5040.

Page 8: 19.2 Permutations and Probability....2016/08/19  · 19.2 Permutations and Probability. 19.3 Combinations and Probability. Use permutations and combinations to compute probabilities

Example 1 Use permutations to find the probabilities.

5 digits 10 digits

10 510!5!

30, 240

all even digits

5 5

5!

0!

120

all even digits

120

30,240

1

252

Page 9: 19.2 Permutations and Probability....2016/08/19  · 19.2 Permutations and Probability. 19.3 Combinations and Probability. Use permutations and combinations to compute probabilities

Explain 2 Finding the Number of Permutations with Repetition

Up to this point, the problems have focused on finding the permutations of distinct objects. If some of the objects are repeated, this will reduce the number of permutations that are distinguishable.

For example, here are the permutations of the letters A, B, and C.

Example 2 Find the number of permutations.A. How many different permutations are there of the letters in the word ARKANSAS.

Answer There are 8 letters in the word, and there 3 A’s and 2 S’s.

So the number of permutations of the letters in the word ARKANSAS is 8!

3!2!= 3360

Permutations with RepetitionThe number of different permutations of n objects where one object repeats 𝑎times, a second object repeats 𝑏 times, and so on is 𝑛!

𝑎! × 𝑏! × ⋯

Page 10: 19.2 Permutations and Probability....2016/08/19  · 19.2 Permutations and Probability. 19.3 Combinations and Probability. Use permutations and combinations to compute probabilities
Page 11: 19.2 Permutations and Probability....2016/08/19  · 19.2 Permutations and Probability. 19.3 Combinations and Probability. Use permutations and combinations to compute probabilities

A Find the probability of getting an alternating boy-girl arrangement.

B Find the probability of getting all of the boys grouped together. The sample space S consists of permutations of __________________, with_________________.

Event A consists of permutations with _______________________ . The possible permutations are BBBBGGGG, GBBBBGGG, n(A) = The probability of getting all the boys grouped together is

Explain 3 Finding a Probability Using Permutations with Repetition (pg. 965)

Example 3 The school jazz band has 4 boys and 4 girls, and they are randomly lined up for a yearbook photo.

8 students 4 boys and 4 girls

all 4 boys in a row

8!4!4!

70

The sample space S consists of permutations of 8 objects, with 4 boys and 4 girls.

Event A consists of permutations that alternate boy-girl or girl-boy. The possible permutations are BGBGBGBG and GBGBGBGB. The probability of getting an alternating boy-girl arrangement is

GGBBBBGG, GGGBBBBG, and GGGGBBBB.𝟓

𝑷 𝑨 =𝒏 𝑨

𝒏 𝑺=

𝟓

𝟕𝟎=

𝟏

𝟏𝟒

(Permutations with repetition can be used to find probablilities.)

Page 12: 19.2 Permutations and Probability....2016/08/19  · 19.2 Permutations and Probability. 19.3 Combinations and Probability. Use permutations and combinations to compute probabilities

What is a combination?A combination is a selection of objects from a group in which order is unimportant.

A restaurant has 8 different appetizers on the menu, as shown in the table. They also offer an appetizer sampler, which contains any 3 of the appetizers served on a single plate. How many different appetizer samplers can be created? The order in which the appetizers are selected does not matter. Nachos Chicken Wings

Chicken Quesadilla Vegetarian Egg Rolls

Potato Skins Soft Pretzels

Beef Chili Guacamole DipA. Find the number appetizer samplers that are possible if the order of selection does matter. This is the number of permutations of 8 objects taken 3 at a time.

B. Find the number of different ways to select a particular group of appetizers. This is the number of permutations of 3 objects.

C. To find the number of possible appetizer samplers if the order of selection does not matter, divide the answer to part A by the answer to part B.

Answer

𝟖!8 3

𝟖!5!

𝟑𝟑𝟔

𝟑!3 3

𝟑!0!

𝟔

So the number of appetizer samplers that can be created is 𝟑𝟑𝟔

𝟔= 𝟓𝟔

Page 13: 19.2 Permutations and Probability....2016/08/19  · 19.2 Permutations and Probability. 19.3 Combinations and Probability. Use permutations and combinations to compute probabilities

Finding a Probability Using Combinations

The number of combinations of the 8 objects taken 3 at a time is

This can be generalized as follows

Here are some other notations

Page 14: 19.2 Permutations and Probability....2016/08/19  · 19.2 Permutations and Probability. 19.3 Combinations and Probability. Use permutations and combinations to compute probabilities

Example 1 Find each probability.

B. There are 52 cards in a standard deck, 13 in each of 4 suits: clubs, diamonds, hearts, and spades. Five cards are randomly drawn from the deck. What is the probability that all five cards are diamonds?

Answer

𝟓𝟐 𝟓

𝟓𝟐!

5!47!𝟐, 𝟓𝟗𝟖, 𝟗𝟔𝟎

𝟓

𝟏𝟑

𝟏𝟑 𝟓

𝟏𝟑!

5!8!𝟏𝟐𝟖𝟕

𝟓

𝟏𝟐𝟖𝟕

2,598,960

𝟑𝟑

66,640

Page 15: 19.2 Permutations and Probability....2016/08/19  · 19.2 Permutations and Probability. 19.3 Combinations and Probability. Use permutations and combinations to compute probabilities

Explain Finding a Probability Using Combinations and Addition

Sometimes, counting problems involve the phrases “at least” or “at most.” For these problems, combinations must be added.

B. Three number cubes are rolled and the result is recorded. What is the probability that at least 2 of the number cubes show 6?

Answer The number of outcomes in the sample space S can be found by using the Fundamental Counting Principle since each roll can result in 1, 2, 3, 4, 5, or 6.

Let A be the event that at least 2 number cubes show 6. This is the sum of 2 events____________________________, or ____________________________________. The event of getting 6 on 2number cubes occurs _____________________since there are __________ possibilities for the other

number cube.

The probability of getting a 6 at least twice in 3 rolls is

𝟔𝟑 𝟐𝟏𝟔

2 number cubes showing 6 3 number cubes showing 65 times 𝟓

Page 16: 19.2 Permutations and Probability....2016/08/19  · 19.2 Permutations and Probability. 19.3 Combinations and Probability. Use permutations and combinations to compute probabilities

Ready for a QUIZ next class?

Classwork and HomeworkPg. 966-971 exercises 1-27 and pg. 978-983 Exercises 1-26