Unit 3 Seminar: Probability and Counting Techniques

Counting(Combinatorics)

I have three shirts: white, blue, pinkAnd two skirts: black, tan

How many different outfits can I make?

Choose shirt

white blue pinkChoose skirt black tan black tan black tan

3*2 = 6 outfits

If you can choose one item from a group of M items and a second item from a group of N items, then the total number of two-item choices you can make is M*N.

A person can be classified by eye color (brown, blue, green), hair color (black, brown, blonde, red) and gender (male, female). How many different classifications are possible?

An ID number consists of a letter followed by 4 digits, the last of which must be 0 or 1. How many different ID numbers are possible?

A permutation is an ordered arrangement of things. For example, the permutations of the word BAD are:

BAD ABD DABBDA ADB DBA

Note: AAA is not a permutation of BAD

We can use the counting principle to count permutations.

Example: How many ways can we arrange the letters GUITAR ?

n! = n(n-1)(n-2) … 1

6! = 6*5*4*3*2*1 = 720

What about repeats?

Example: How many ways can we arrange the letters MISSISSIPPI ?

Sometimes we don’t use all of the available items.

Example: How many ways can we arrange three of the letters WINDY ?

“permutations of size 3, taken from 5 things”

How many ways can a President, Vice President and Secretary be chosen from a group of 10 people?

How many selections of 2 letters from the letters WIND can be made (order doesn’t matter) ?

The number of combinations of n things taken r at a time:

How many ways can three people be chosen from a group of 10 people?

Basic Probability

1.) Classical – based on theoryex: games of chance

2.) Empirical – based on historical observations ex: sports betting

3.) Subjective – based on an educated guess or a rational belief in the truth or falsity of propositionssee: “A Treatise on Probability” by John Maynard Keynes

EXPERIMENT: Throw a single die.

Sample Space S = {1,2,3,4,5,6}An event is a subset of the sample space

Ex: throw an even number E = {2,4,6}

The probability of an event

P(E) = n(E)/n(S) = 3/6 = 1/2

Select a card from a deck of 52 cards.What is the probability that it is:

1.) an ace2.) the jack of clubs3.) not a queen4.) the king of stars5.) a heart, diamond, club or spade

A dartboard has the shape shown.

What is P(7) ?

23

4

7

1 5 6

Prof. Smith’s grades for a course in College Algebra over three years are:

A = 40B = 180C = 250D = 90F = 60

If Jane takes his course, what is the probability that she will get a C or better?

Odds in favor of an event = P(success) / P(failure)

= P(it happens) / P(it doesn’t happen)

Ex. A coin is weighted so that P(heads) = 2/3. What are the odds of getting heads?

What are the odds of rolling a 4 with a fair die?

The probability of rain today is .35. What are the odds in favor of rain today?

Expected Value 

The average result that would be obtained if an experiment were repeated many times.

Suppose you have as possible outcomes of the experiment events A1 , A2 , A3 with probabilities P1 , P2 , P3

Expected Value = P1* A1 + P2 *A2 + P3 * A3

An investment club is considering buying a certain stock. Research shows that there is a 60% chance of making $10,000, a 10% chance of breaking even, and a 30% chance of losing $7200.

Determine the expected value of this purchase.

Game: Blindfolded, throw a dart. What is the expectation?

$5

$1 $10 $20

$50