lesson 10.5. the numerator and denominator of a theoretical probability are numbers of...

Lesson 10.5

The numerator and denominator of a theoretical probability are numbers of possibilities.

Sometimes those possibilities follow regular patterns that allow you to “count” them.

The numerator and denominator of a theoretical probability are numbers of possibilities.

Sometimes those possibilities follow regular patterns that allow you to “count” them.

Suppose you want to create a random playlist from a library of songs on an MP3 player. If you do not repeat any songs, in how many different orders do you think the songs could be played?

In this investigation you will discover a pattern allowing you to determine the number of possible orders without listing all of them.

Step 1 Start by investigating some simple cases. Consider libraries of up to five songs, and playlists of up to five of those songs. ◦ Notice n represents the number of songs in the

library (1≤n≤5) and r represents the length of the playlist (1≤r≤n).

Step 1 Start by investigating some simple cases. Consider libraries of up to five songs, and playlists of up to five of those songs. ◦ Suppose we wanted to make playlists of two songs

from a list of three songs. Let A, B, and C represent the three songs available.

◦ Write out the all the playlists that could be formed. ◦ AB, AC, BA, BC, CA, and CB. ◦ How do you know you have all the ways listed?

6

Start with n = 1, 2, and 3 and write out all the playlists you can form

Describe any patterns you found in either the rows or columns of the table.

1 2 3

2

6

Have you used a tree diagram?You have

• four choices of sweaters• six different pants• two pairs of shoes

How many different outfits are there? Why do we multiply?

We can think about using the Counting Principal

We are going to put outfits together with a sweater, a pair of pants and a pair of shoes. A box could be used for each item

How many choices do I have for each box?

4 X X =324 2

Why should I multiply these numbers?

1 2 3 4 5

62 12 20

6 24 60

24 120

120

Describe any patterns you found in either the rows or columns of the table.

Finish the chart for the number of playlists

Write an expression for the number of ways to arrange 10 songs in a playlist from a library of 150 songs.

150x149x148x147x146x145x144x143x142x141

Suppose there are n1 ways to make a choice, and for each of these there are n2 ways to make a second choice, and for each of these there are n3 ways to make a third choice, etc. The product of

gives the number of possible outcomes.

n1 ● n2● n3● …..

Suppose a set of license plates has any three letters from the alphabet, followed by any three digits. ◦ How many different license plates are

possible?◦ What is the probability that a license

plate has no repeated letters or numbers?

___ ___ ___ ___ ___ ___26 26 26 10 10 10

17, 576, 000 license plates

◦ What is the probability that a license plate has no repeated letters or numbers?

__ __ __ __ __ __

About 63.9% of license plates would not have repeated letters or numbers.

26 25 24 10 9 8 =11,232,000

When the objects cannot be used more than once, the number of possibilities decreases at each step. These are called “arrangements without replacement.” This arrangement is called a permutation.

nPr is the number of permutations of n things chosen r at a time.

Seven flute players are performing in an ensemble.◦ The names of all seven players are listed in

the program in random order. What is the probability that the names are in alphabetical order?

◦ After the performance, the players are backstage. There is a bench with room for only four to sit. How many possible seating arrangements are there?

◦ What is the probability that the group of four players is sitting in alphabetical order?

Seven flute players are performing in an ensemble.◦ The names of all seven players are listed in the program

in random order. What is the probability that the names are in alphabetical order?

7P7 =7 •6 • 5 • 4 • 3 • 2 • 1= 5040

Only one of these arrangements is in alphabetical order. The probability is 1/5040 or approximately 0.0002.

Seven flute players are performing in an ensemble.◦ After the performance, the players are backstage. There

is a bench with room for only four to sit. How many possible seating arrangements are there?

7P4 =7•6•5•4=840

Seven flute players are performing in an ensemble.◦ What is the probability that the group of

four players is sitting in alphabetical order?

With each arrangement of 4, there is only one correct order.

1/24, or approximately 0.04167.

Notice that the answer to part c does not depend on the answer to part b

4P4 =4• 3 • 2 • 1 = 24

!( )!n r

nP

n r