Sign in to wikis by wetpaint

MDM4U1@FMG

Home Discussions Photos Videos News Updates Members

Sign in or

Report page Share Share this

6.3 Probabilities Using Counting TechniquesThis is a featured page
In a number of different situations, it is not easy to determine the outcomes of an event by counting them individually. Alternatively, counting techniques that involve permutations and combinations are helpful when calculating theoretical probabilities.

This section will examine methods for determining theoretical probabilities of successive or multiple events.

Permutation? or Combination?

The following flow chart will help determine which formula is suitable for any given question. By simply following a series of "yes" or "no" questions, the appropriate formula can be determined.
Flow

Ex. 1 - Using Permutations:

6.3 Probabilities Using Counting Techniques - MDM4U1@FMG
6.3 Probabilities Using Counting Techniques - MDM4U1@FMG
The specific outcome of Mike starting in lane 1 and the other two starting in lane 2 and lane
3 can only happen one way, so n(A) = 1. Therefore,

6.3 Probabilities Using Counting Techniques - MDM4U1@FMG
The probability that Mike will start in the first lane next to his other brothers in lane 2 and 3 is approximately 0.00101.

Ex. 1(a) - Using Permutations:

Exactly Three People form a line at a grocery store. What is the probability that they will line up in descending order of age? (I.e. oldest, middle and youngest)

→Solution using the blank like method:

n(A): # of ways they will line up in descending order of age, thus:

6.3 Probabilities Using Counting Techniques - MDM4U1@FMG

n(S): # of ways all three can line up in any order, thus:

6.3 Probabilities Using Counting Techniques - MDM4U1@FMG

To calculate the probability that the three people will line up in descending order is:

6.3 Probabilities Using Counting Techniques - MDM4U1@FMG

Ex. 2 - Using Combinations:

6.3 Probabilities Using Counting Techniques - MDM4U1@FMG
6.3 Probabilities Using Counting Techniques - MDM4U1@FMG
However, the group may consist of any 3 members from the group of 13. Thus, n(S) is the total number of possible outcomes.

6.3 Probabilities Using Counting Techniques - MDM4U1@FMG
The probability of selecting a group consisting of runners only is:

6.3 Probabilities Using Counting Techniques - MDM4U1@FMG
Therefore, the probability of selecting a committee with 3 runners is approximately 0.03497.

b) Either the committee is comprised of runners only, or it isn't. Therefore, the probability of the complement A, P(A'), gives the desired result.

6.3 Probabilities Using Counting Techniques - MDM4U1@FMG
So the probability of selecting a committee not comprised of runners is approximately 0.96503.

Ex. 2(b) – Using Combinations:

A group of 12 people is going out town on Saturday night. The group will take three cars with four people in each car. If they distribute themselves among the cars at random, what is the probability that Chantal and Rafael will be in the same car? (*note: f#: friend)

→Solution using the blank line method:

n(A):# of ways Chantal and Rafael will be in the same car, thus:

6.3 Probabilities Using Counting Techniques - MDM4U1@FMG n(S):# of ways Chantal and Rafael will be in any one of the three cars, thus:

6.3 Probabilities Using Counting Techniques - MDM4U1@FMG

6.3 Probabilities Using Counting Techniques - MDM4U1@FMG
6.3 Probabilities Using Counting Techniques - MDM4U1@FMG

Ex. 3 - Using the Fundamental Counting Principle:
6.3 Probabilities Using Counting Techniques - MDM4U1@FMG
6.3 Probabilities Using Counting Techniques - MDM4U1@FMG
For the third employee, since there are 363 ways this person can have a different birthday from the other two, the probability that all three employees have different birthdays is:
6.3 Probabilities Using Counting Techniques - MDM4U1@FMG
Now continue the process with the 12 people in the office:
6.3 Probabilities Using Counting Techniques - MDM4U1@FMG
The probability that at least two people in the office have the same birthday is approximately
0.167.

Key Concepts:

In probability experiments with many possible outcomes, you can apply the fundamental

counting principle and techniques using permutations and combinations.

Permutations are useful when order is important in the outcomes; combinations are

useful when order is not important.

References:

“Gliffy: BETA.” Gliffy. San Francisco, California, 2005. 18 Dec 2006.

.

Wgman, Diane. Mathematics of Data Management. Toronto: McGraw-Hill Ryerson, 2002.

Humza_89
Humza_89 Latest page update: made by Humza_89 , May 18 2007, 9:49 PM EDT (about this update - complete history)
Keyword tags: combination counting technique permutation Probability
More Info: links to this page
Share this Twitter Facebook MySpace Stumble Upon Delicious Add to Favorites Email
Threads for this page Started By Thread Subject Replies Last Post Anonymous .... 0 Mar 12 2013, 10:18 PM EDT by Anonymous

Thread started: Mar 12 2013, 10:18 PM EDT Watch
The permutation part of the first example should actually be 11P8 shouldn't it?
Do you find this valuable?

Post reply

Anonymous PCI students 0 Nov 12 2012, 10:05 PM EST by Anonymous Anonymous Park City Independent 0 Nov 12 2012, 9:56 PM EST by Anonymous Start a New Thread as Anonymous (Get credit for your thread) Subject:
Message:
Tags:
Category:
Showing 3 of 11 threads for this page - view all
Related Content (what's this?)
.... PCI students Park City Independent
Hi! Hello math its easy O.o alia
Hey 6.1 Basic Probability Concepts 7.2 Binomial Distributions

Home 1.1 Iterative Process 1.5 Graph Theory 1.6 Matrices Algebra 1.7 Problem solving with Matrices 2.5 Measures of Central Tendency (Mean, Median and Mode) 3.1 Scatter Plots and Linear Correlation 3.3 Non-linear regression 4.1 Organized Counting 4.2 Factorials and Permutations 4.3 Permutations with Some Identical Elements 4.4 Pascals Triangle 4.5 Applying Pascals Methods 5.1 Organized Counting with Venn Diagrams 5.2 Combinations... 5.3 Problem Solving With Combinations (Part 1) 5.4 The Binomial Theorem 6.1 Basic Probability Concepts 6.2 Odds 6.3 Probabilities Using Counting Techniques 6.4 Independent and Dependent events 6.5 Mutually Exclusive Events 6.6 Applying Matrices to Probability Problems 7.1 Probability Distributions 7.2 Binomial Distributions 7.3 Geometric Distribution 7.4 Hypergeometric Distribution Linear Regression (good example) Linear Regression (mediocre example) Linear Regression (poor example) Tests

start a free website

Wikis on Wetpaint Contact Us Help Sitemap Privacy Terms of Service

Featured Sites Glee American Idol The Bachelor Pretty Little Liars The Vampire Diaries Gossip Girl Grey's Anatomy Bones The Bachelorette Real Housewives of Beverly Hills America's Next Top Model Real Housewives of Atlanta Jersey Shore Kourtney & Kim Take New York Dancing With the Stars

Site Meter