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Unit 3 Seminar:
Probability and Counting Techniques
Counting
(Combinatorics)
I have three shirts: white, blue, pink
And two skirts: black, tan
How many different outfits can I make?
Choose shirt
white
Choose skirt
black
tan
blue
black tan
3*2 = 6 outfits
pink
black tan
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
BDA
ABD
ADB
DAB
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.)
2.)
3.)
Classical – based on theory
ex: games of chance
Empirical – based on historical
observations
ex: sports betting
Subjective – based on an educated guess
or a rational belief in the truth
or falsity of propositions
see: “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.)
2.)
3.)
4.)
5.)
an ace
the jack of clubs
not a queen
the king of stars
a heart, diamond, club or spade
A dartboard has the shape shown.
2
7
4
What is P(7) ?
1
3
5
6
Prof. Smith’s grades for a course in College
Algebra over three years are:
A = 40
B = 180
C = 250
D = 90
F = 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