Copyright © 2011 Pearson Education, Inc. Probability: Living with the Odds.
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Transcript of Copyright © 2011 Pearson Education, Inc. Probability: Living with the Odds.
Copyright © 2011 Pearson Education, Inc. Slide 7-4
7-E
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 © 2011 Pearson Education, Inc. Slide 7-5
7-E
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 © 2011 Pearson Education, Inc. Slide 7-6
7-E
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 © 2011 Pearson Education, Inc. Slide 7-7
7-E
The Permutations FormulaExample: If 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 © 2011 Pearson Education, Inc. Slide 7-8
7-E
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 © 2011 Pearson Education, Inc. Slide 7-9
7-E
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 © 2011 Pearson Education, Inc. Slide 7-10
7-E
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 © 2011 Pearson Education, Inc. Slide 7-11
7-E
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 © 2011 Pearson Education, Inc. Slide 7-12
7-E
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%!
Copyright © 2011 Pearson Education, Inc. Slide 7-13
7-E
Birthday Coincidence
x = People in Room: 10 15 20 25 30 35 40 45 y = Probabilities: .117 .253 .411 .569 .706 .814 .891 .940
1
0y =
Pro
babi
litie
s
x = People in Room
What are the probabilities that someone in a room of x people will have a birthday in common with someone else in that room?
Copyright © 2011 Pearson Education, Inc. Slide 7-14
7-E
Birthday Coincidence
x = People in Room: 10 15 20 25 30 35 40 45 y = Probabilities: .117 .253 .411 .569 .706 .814 .891 .940
1
0y =
Pro
babi
litie
s
x = People in Room
What are the probabilities that someone in a room of x people will have a birthday in common with someone else in that room?
Copyright © 2011 Pearson Education, Inc. Slide 7-15
7-E
Birthday Coincidence
x = People in Room: 10 15 20 25 30 35 40 45 y = Probabilities: .117 .253 .411 .569 .706 .814 .891 .940
1
0y =
Pro
babi
litie
s
x = People in Room
What are the probabilities that someone in a room of x people will have a birthday in common with someone else in that room?
Copyright © 2011 Pearson Education, Inc. Slide 7-16
7-E
Birthday Coincidence
x = People in Room: 10 15 20 25 30 35 40 45 y = Probabilities: .117 .253 .411 .569 .706 .814 .891 .940
1
0y =
Pro
babi
litie
s
x = People in Room
What are the probabilities that someone in a room of x people will have a birthday in common with someone else in that room?
Copyright © 2011 Pearson Education, Inc. Slide 7-17
7-E
Birthday Coincidence
x = People in Room: 10 15 20 25 30 35 40 45 y = Probabilities: .117 .253 .411 .569 .706 .814 .891 .940
1
0y =
Pro
babi
litie
s
x = People in Room
What are the probabilities that someone in a room of x people will have a birthday in common with someone else in that room?
Copyright © 2011 Pearson Education, Inc. Slide 7-18
7-E
Birthday Coincidence
x = People in Room: 10 15 20 25 30 35 40 45 y = Probabilities: .117 .253 .411 .569 .706 .814 .891 .940
1
0y =
Pro
babi
litie
s
x = People in Room
How many people in the room would be required for 100% certainty?
What are the probabilities that someone in a room of x people will have a birthday in common with someone else in that room?