Probability
Module 8: Probability Counting Principles
Permutations
Lesson
Factorials
Factorial notation is indicated by n!, and is equal to the product of all the positiv e integers less than or equal to n. 0! is alway s equal to 1.
For example, 5! = 5 • 4 • 3 • 2 •1 = 120.
Using f actorials to count the number of arrangments is an application of the counting principle. The number of way s that n items can be arranged is equal to n!
Look f or the f actorial button on y our calculator.
Example:
6 runners are in a race. In how many way s can they f inish? Solution:
In the Counting Principles lesson, y ou learned that there are 6 possible outcomes f or f irst place. Once that place has been assigned, there are 5 possible choices f or 5th place, 4 choices f or 4th place and so on. The number of possible outcomes is 6 • 5 • 4 • 3 • 2 • 1 = 720. This is the same as 6! There are 6! way s to arrange 6 items.
Permutations
A permutation is an ordering of objects. The number of permutations of n objects taken r at a time can be f ound using the f ollowing f ormula: It is also
deriv ed f rom the counting principles that were studied earlier in this course.
Example:
A swim team has 12 swimmers who can swim in a f reesty le ev ent. The swim coach will choose 4 of these swimmers to go f irst, second, third, and f ourth. In how many dif f erent way s can this be done? To determine the number of dif f erent swimmer orders that are possible, use the f ormula nPr with n = 12 and r = 4.
Solution:
12P4 =
There are 11,880 dif f erent possible swimmer orders the coach can choose f rom the relay race.
This could also be done using counting princeiples, there are 12 choices f or the f irst swimmer, 11 f or the third, and so on...
Example:
There are 10 empty desks in a room. In how many way s can a teacher assign 5 students to the 10 desks? Since order matters, use the f ormula 10P5.
Solution:
10P5 =
Arrangements of Letters Without Repetition
Example:
How many distinguishable perutations are possible using the letters in the word: TEXAS?
Solution:
There are 5 letters in the word. The number of permutations is .
Arrangements of Letters With Repetition
Example:
How many distinguishable permutations are possible using the the letters in the word: BANANA?
Solution:
In this case, letters are repeated. Since the letters N and A appear more than once, some of the permutations will be identical. To account f or these, div ide the total number of permutations, 6!, by the f actorial of the number of times duplicate letters appear. Since A appears three times, div ide by 3!. Since N appears twice, div ide by 2!
The total number of arrangements is
Probability
Probability is the number of f av orable outcomes div ided by the number of possible outcomes. You may need to use permutations to count the number of outcomes. Additionally, y ou may need to use the multiplication and/or addition rules to f igure probabilities with multiple outcomes.
Example:
11 school buses are randomly lined up outside the school. What is the probability that bus #1201 will be f irst and bus #1152 will appear last?
Solution:
Number of f av orable outcomes: There is exactly 1 choice f or bus #1201 to be f irst, and exactly one choice f or bus #1152 to be last. The 9 buses in the middle can be arranged in 9! dif f erent way s.
Number of f av orable outcomes = 1 • 9! • 1 = 9!
The number of possible outcomes: 11 objects can be arranged in 11! way s.
P(bus #1201 f irst and bus #1152 last) =
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Example:
There are 10 play ers on a sof tball team. Each game the batting order is chosen randomly. Find the probability that y ou are chosen to bat f irst, and y our bet f riend is chosen second.
Solution:
Probability is the quotient of th enumber of f av orable outcomes and the number of possible outcomes.
The number of f av orable outcomes: There is one choice f or y ou f irst, and one choice f or y our f riend second. That leav es 8! f or the remaining 8 slots.
The number of possible outcomes: 10!
Self-Check
Self-Check Problem 1
In how many way s can y ou arrange 4 of the letters in the word VIRTUAL?
C heck A nswer
Self-Check Problem 2
Your band has written 11 songs and plans to record 5 of them f or a CD. In how many way s can y ou arrange the songs on the CD?
a. 55
b. 120
c. 55,440
d. 39,916,800
C heck A nswer
Self-Check Problem 3
You are in a soapbox racing competition. In each heat, 9 cars race and the positions of the cars are randomly assigned. What is the probability that y ou are chosen to be in the
f irst or second position of the heat in which y ou are racing?
C heck A nswer
Permutations https://cobbk12.blackboard.com/bbcswebdav/institution/eHigh School/C...
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