Copyright © 2011 Pearson Education, Inc. Probability: Living with the Odds.

Copyright © 2011 Pearson Education, Inc.

Probability: Living with the Odds

Copyright © 2011 Pearson Education, Inc. Slide 7-3

Unit 7E

Counting and Probability


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.


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


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


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


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


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


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.


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


7-E

The probability that at least two people in a class of 25 have the same birthday is


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%!


7-E


x = People in Room: 10 15 20 25 30 35 40 45 y = Probabilities: .117 .253 .411 .569 .706 .814 .891 .940

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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?


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x = People in Room



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1

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x = People in Room

How many people in the room would be required for 100% certainty?


Copyright © 2011 Pearson Education, Inc. Probability: Living with the Odds.

Documents

room of x people

possible committees

possible candidates

group of n items

possible relay teams

ebirthday coincidencewhat

different birthdays

group of n choices