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

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