### 4.2 Probability Models

```4.1 (cont.) Probability Models
The Equally Likely Approach
(also called the Classical
Approach)
Assigning Probabilities
If an experiment has N outcomes, then
each outcome has probability 1/N of
occurring
If an event A1 has n1 outcomes, then
P(A1) = n1/N
Dice
You toss two dice. What is the probability of the outcomes summing to 5?
This is S:
{(1,1), (1,2), (1,3),
……etc.}
There are 36 possible outcomes in S, all equally likely (given fair dice).
Thus, the probability of any one of them is 1/36.
P(the roll of two dice sums to 5) =
P(1,4) + P(2,3) + P(3,2) + P(4,1) = 4 / 36 = 0.111
We Need Efficient Methods
for Counting Outcomes
Product Rule for Ordered
Pairs
A student wishes to commute to a junior
college for 2 years and then commute to a
state college for 2 years. Within
commuting distance there are 4 junior
colleges and 3 state colleges. How many
junior college-state college pairs are
available to her?
Product Rule for Ordered
Pairs
junior colleges: 1, 2, 3, 4
state colleges a, b, c
possible pairs:
(1, a) (1, b) (1, c)
(2, a) (2, b) (2, c)
(3, a) (3, b) (3, c)
(4, a) (4, b) (4, c)
Product Rule for Ordered
Pairs
junior colleges: 1, 2, 3, 4
state colleges a, b, c
4 junior colleges
3 state colleges
possible pairs:
total number of possible
(1, a) (1, b) (1, c)
pairs = 4 x 3 = 12
(2, a) (2, b) (2, c)
(3, a) (3, b) (3, c)
(4, a) (4, b) (4, c)
Product Rule for Ordered
Pairs
junior colleges: 1, In
2,general,
3, 4 if there are n1 ways
to choose the first element of
state colleges a, b,thec pair, and n ways to choose
2
the second element, then the
possible pairs:
number of possible pairs is
(1, a) (1, b) (1, c) n1n2. Here n1 = 4, n2 = 3.
(2, a) (2, b) (2, c)
(3, a) (3, b) (3, c)
(4, a) (4, b) (4, c)
Counting in “Either-Or” Situations
• NCAA Basketball Tournament, 68 teams:
how many ways can the “bracket” be
filled out?
1. How many games?
2. 2 choices for each game
3. Number of ways to fill out the bracket:
267 = 1.5 × 1020
•
•
Earth pop. about 6 billion; everyone fills
out 100 million different brackets
Chances of getting all games correct is
begins with a number from 1 to 26,
corresponding to the 26 counties in a
state. This number is followed by a 5digit number. How many different
license plates can the state issue?
1.
2.
3.
4.
5.
1,300
6,552
2,600,000
786,240
26,000
0%
1
0%
0%
2
3
0%
0%
4
5
10
Counting Example
rearranging the order of the questions on
a survey
If Gallup has a 5-question survey, how
many different versions of the survey are
required if all possible arrangements of
the questions are included?
Solution
There are 5 possible choices for the first
question, 4 remaining questions for the
second question, 3 choices for the third
question, 2 choices for the fourth
question, and 1 choice for the fifth
question.
The number of possible arrangements is
therefore
5  4  3  2  1 = 120
Efficient Methods for
Counting Outcomes
Factorial Notation:
n!=12 … n
Examples
1!=1; 2!=12=2; 3!= 123=6; 4!=24;
5!=120;
Special definition: 0!=1
Factorials with calculators
and Excel
Calculator:
non-graphing: x ! (second function)
graphing: bottom p. 9 T I Calculator
Commands
(math button)
Excel:
Insert function: Math and Trig category,
FACT function
Factorial Examples
20! = 2.43 x 1018
1,000,000 seconds?
1,000,000,000 seconds?
31 years = 109 seconds
1018 = 109 x 109
20! is roughly the age of the universe in
seconds
Permutations
A B C D E
How many ways can we choose 2 letters
from the above 5, without replacement,
when the order in which we choose the
letters is important?
5  4 = 20
Permutations (cont.)
5!
5!
5  4  20 
  5 4
(5  2)! 3!
5!
Notation: 5 P2 
 20
(5  2)!
Permutations with
calculator and Excel
Calculator
non-graphing: nPr
Graphing
p. 9 of T I Calculator Commands
(math button)
Excel
Insert function: Statistical, Permut
Combinations
A B C D E
How many ways can we choose 2 letters
from the above 5, without replacement,
when the order in which we choose the
letters is not important?
5  4 = 20 when order important
Divide by 2: (5  4)/2 = 10 ways
Combinations (cont.)
 
5!
5! 5  4 20
 5 C2 



10
(5  2)!2! 3!2! 1 2 2
 
n!
 n Cr 
(n  r )!r!
5
2
n
r
BUS/ST 350 Powerball Lottery
From the numbers 1 through 20,
choose 6 different numbers.
Write them on a piece of paper.
Chances of Winning?
Choose6 numbersfrom 20, without
replacement, order not important.
Number of possibilities?
 
20
6
20!
 20 C6 
 38,760
(20  6)!6!
Example: Illinois State
Lottery
Choose6 numbersfrom54 numbers wit hout
replacement ; order not import ant
54!
 25,827,165
54 C6 
48!6!
(about 1 secondin 10 mont hs)
(1200ft 2 house,16.5million ping pongballs)
North Carolina Powerball
Lottery
Prior to Jan. 1, 2009
After Jan. 1, 2009
5 from 1 - 55:
5 from 1 - 59:
55!
 3, 478, 761
5!50!
59!
 5, 006, 386
5!54!
1 from 1 - 42 (p'ball #):
1 from 1 - 39 (p'ball #):
42!
 42
1!41!
39!
 39
1!38!
3, 478, 761*42 
5, 006, 386*39 
146,107, 962
195, 249, 054
Most recent change: powerball number is from 1 to 35
http://www.nc-educationlottery.org/faq_powerball.aspx#43
The Forrest Gump Visualization of
How large is 195,249,054?
\$1 bill and \$100 bill both 6” in length
10,560 bills = 1 mile
one \$100 bill …
… and take a long walk, putting down bills
end-to-end as we go
Raleigh to Ft. Lauderdale…
… still plenty of bills
remaining, so continue
from …
… Ft. Lauderdale to San
Diego
… still plenty of bills remaining, so continue from…
… San Diego to Seattle
… still plenty of bills remaining, so continue from …
… Seattle to New York
… still plenty of bills remaining, so continue from …
… New York back to
Raleigh
… still plenty of bills remaining, so …
Go around again! Lay a
second path of bills
Still have ~ 5,000 bills left!!
Chances of Winning NC
Powerball Lottery?
Remember: one of the bills you put down
is a \$100 bill; all others are \$1 bills.
Put on a blindfold and begin walking
along the trail of bills.
Your chance of winning the lottery: the
chance of selecting the \$100 bill if you
stop at a random location along the trail
and pick up a bill .
More Changes
After Jan. 1, 2009
5 from 1 - 59:
59!
 5, 006,386
5!54!
1 from 1 - 39 (p'ball #):
39!
 39
1!38!
5, 006,386*39 
195, 249, 054
After Jan. 1, 2012
 http://www.nceducationlottery.org/pow
erball_how-to-play.aspx
Virginia State Lottery
50!
P ick 5 : 50 C5 
 2,118,760
45!5!
2,118,760  25 C1 
25!
2,118,760
 52,969000
24!1!
```