copyright © 2015, 2011, 2008 pearson education, inc. chapter 7, unit e, slide 1 probability: living...
TRANSCRIPT
Copyright © 2015, 2011, 2008 Pearson Education, Inc. Chapter 7, Unit E, Slide 1
Probability: Living With The Odds
7
Copyright © 2015, 2011, 2008 Pearson Education, Inc. Chapter 7, Unit E, Slide 2
Unit 7E
Counting and Probability
Copyright © 2015, 2011, 2008 Pearson Education, Inc. Chapter 7, Unit E, Slide 3
If we make r selections from a group of n choices, a total of different arrangements are possible.
Example: How many 7-number license plates are possible?
Arrangements with Repetition
rnnnn
71010101010101010
There are 10 million different possible license plates.
Copyright © 2015, 2011, 2008 Pearson Education, Inc. Chapter 7, Unit E, Slide 4
Permutations
We are dealing with permutations whenever
all selections come from a single group of items, no item may be selected more than once, and the order of arrangement matters.
e.g., ABCD is different from DCBA
The total number of permutations possible with a group of n items is n!, where
121! nnn
Copyright © 2015, 2011, 2008 Pearson Education, Inc. Chapter 7, Unit E, Slide 5
Example
A middle school principal needs to schedule six different classes—algebra, English, history, Spanish, science, and gym—in six different time periods. How many different class schedules are possible?
Solution
6! = 6(5)(4)(3)(2)(1) = 720
The principal can schedule the six classes in 720 different ways
Copyright © 2015, 2011, 2008 Pearson Education, Inc. Chapter 7, Unit E, Slide 6
If we make r selections from a group of n choices, the number of permutations (arrangements in which order matters) is
The Permutations Formula
!1 2 1
!n r
nP n n n n r
n r
Example: On a team of 10 swimmers, how many possible 4-person relay teams are there?
There are possible relay teams!
504078910
Copyright © 2015, 2011, 2008 Pearson Education, Inc. Chapter 7, Unit E, Slide 7
ExampleIf an international track event has 8 athletes participating and three medals (gold, silver and bronze) are to be awarded, how many different orderings of the top three athletes are possible?
There are 336 different orderings of the top three athletes!
8 3
8! 8 7 6 5!8 7 6 336
8 3 ! 5!P
Copyright © 2015, 2011, 2008 Pearson Education, Inc. Chapter 7, Unit E, Slide 8
Example
A Little League manager has 15 children on her team. How many ways can she form a 9-player batting order?
Solution
Nearly 2 billion batting orders are possible for a baseball team with a roster of 15 players.
15 9
15! 15!1,816,214,400
(15 9)! 6!
P
Copyright © 2015, 2011, 2008 Pearson Education, Inc. Chapter 7, Unit E, Slide 9
Combinations
Combinations occur whenever
all selections come from a single group of items, no item may be selected more than once, and the order of arrangement does not matter
e.g., ABCD is considered the same as DCBA
If we make r selections from a group of n items, the number of possible combinations is
!!
!
! rrn
n
r
PC rnrn
Copyright © 2015, 2011, 2008 Pearson Education, Inc. Chapter 7, Unit E, Slide 10
Example: If a committee of 3 people are needed out of 8 possible candidates and there is not any distinction between committee members, how many possible committees would there be?
The Combinations Formula
There are 56 possible committees!
56123
678
!3!5
!5678
!3!5
!8
!3!38
!838
C
Copyright © 2015, 2011, 2008 Pearson Education, Inc. Chapter 7, Unit E, Slide 11
Probability and Coincidence
Example: What is the probability that at least two people in a class of 25 have the same birthday?
The answer has the form
Although a particular outcome may be highly unlikely, some similar outcome may be extremely likely or even certain to occur.
Coincidences are bound to happen.
Copyright © 2015, 2011, 2008 Pearson Education, Inc. Chapter 7, Unit E, Slide 12
Birthday Coincidence
24
364 363 341 364 363 341
365 365 365 365
The probability that all 25 students have different birthdays is
61
61
1.348 100.431
3.126 10
Copyright © 2015, 2011, 2008 Pearson Education, Inc. Chapter 7, Unit E, Slide 13
The probability that at least two people in a class of 25 have the same birthday is
Birthday Coincidence
P(at least one pair of shared birthdays)
= 1 – P(no shared birthdays)
≈ 1 – 0.431 ≈ 0.569 ≈ 57%
The probability that at least two people in a class of 25 have the same birthday is approximately 57%!