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Probability
EQ: What’s the chance you’re going to understand
this?
Deal or No Deal
 In the game show Deal or No Deal, contestants play and deal for up to
$1,000,000. By collecting and analyzing data, you can determine the
chances of winning $1,000,000.
 Each game consists of 26 briefcases containing dollar amounts ranging
from $.01 to $1,000,000. Without knowing the amount in each
briefcase, the contestant chooses one briefcase, which remains sealed
until the end of the game. In each round, a predetermined number of
the remaining briefcases are opened revealing the amount in each. At
the end of each round, the “Banker” offers the contestant an amount of
cash based on the amounts still left in the unopened briefcases, in
exchange for the contestant’s briefcase. The contestant can either accept
the Banker’s offer and end the game or continue to the next round. If
the contestant never accepts an offer an all other briefcases have been
opened, then the contestant receives what is in the briefcase he or she
chose at the beginning of the game.
Basic Questions:
What is the probability that the first briefcase chosen contains
$1,000,000?
1
 .0385  3.85%
26
If seven of the briefcases have at least $100,000,
what is the probability of wining at least $100,000?
7
 .2692  26.92%
26
What is the probability of a contestant choosing a briefcase
with less than $100,000?
26  7 19

 .7308  73.08%
26
26
Sooo… What is the whole idea?
Law of large numbers (LLN):
As the number of independent trials increases, the long-run relative
frequency of repeated events gets closer and closer to a single
value.
To understand the “big idea” we are going to take a
look at similulations.
Ex: Simulate flipping a coin
Using random numbers we can flip an imaginary coin:
1. Define the events?
Heads or Tails
2. Assign random numbers to events?
Using digits 0 to 9:
Using digits 1 and 2:
0-4 = Heads 5-9 = Tails
Even = Heads Odd = Tails
1=Heads
3. Decide on how many trials.
4. Execute simulation
2 = Tails
Results of simulation
Trial 1: {1111211121} Results: P(T) = 2/10 P(H)=8/10
Trial 2: {2212211221} Results: P(T) = 6/10 P(H)=4/10
Trial 3: {1222112111}
Results: P(T) = 4/10 P(H)=6/10
Combined Results: P(T) = 12/30 P(H) = 18/30
Chance behavior is RANDOM in the short run
Empirical Probability: the probability resulting
from observing a phenomenon
Simulation vs. Reality
Theoretical probability: mathematical model of random
phenomenon
Theoretical probability of flipping a coin:
P( Heads ) 
event of interest 1

possible events
2
According to LLN, as the number of trials increase the
empirical probability will equal the theoretical
probability.
Probability models
Terminology
Trial: a random event
Outcome: the value of the random event
Sample space: the set of all possible outcomes
Sample Spaces
List the sample space for each of the following.
1. Flip 3 coins, one after the other.
S={HHH, HHT, HTT, HTH, THH, THT, TTH, TTT}
2. Roll two dice; record the sum of the numbers showing
S={2,3,4,5,6,7,8,9,10,11,12}
Creating Probability Models
Why are they useful????
To determine is something is fair!!! (ie equally likely)
We can create the model by determining the number of ways you can
achieve each outcome.
Creating Probability Models
Let’s take a look at our last example…
Determine the number of way we can achieve each outcome.
Event
Possible outcomes to achieve event
Sum of the
outcomes
Creating Probability Models
Now, let’s find the probability that each will occur.
Determine the number of way we can achieve each outcome.
Event
Probability
What do you notice about the sum of the probabilities above?
Probability models
Properties
0  P( A)  1
P (Sample Space)  1
The probability of an event is between 0 and 1
The set of all outcomes of a trial must have probability 1
P( A)  1  P( AC )
or
P( A)  1  P( A )
The Probability of an event occurring is 1 minus the
Probability that it does not occur.
P( A  B)  P( A or B)  P( A)  P( B)
P( A  B)  P( A and B)  P( A) * P( B)
If events are disjoint
If events are independent
Example
 In 2001 M&M’s decided to add another color to the standard
color lineup of brown, yellow, red orange, blue and green.
To decide which color to add, they surveyed people in nearly
every country of the world and asked them to vote among
purple pink and teal. The winner was purple.
In the U.S. 42% of those who voted said purple, 37% said teal, and
only 19% said pink. In Japan the percentages were 38% pink, 36%
teal, and only 16% purple. Using Japan’s percentages answer the
following questions.
List the sample space for the random event of choosing a color
Verify that the first two properties are true.
0  P( A)  1
P (Sample Space)  1
Identify a new event for no preference (none) and
assign it a probability.
P(pink)=.38
P(teal)=.36
P(purple)=.16 P(none) =.10
1. What is the probability that a Japanese survey respondent
selected at random preferred either pink or teal?
P(pink or teal )  P( Pink )  P(teal )  .38  .36  .74
2. If we pick two respondents at random, what is the
Probability that they both said purple?
P(purple and purple )  P( purple ) * P( purple )  (.16)(. 16)  .0256
3. If we pick three respondents at random, what is the
Probability that at least one preferred purple?
P(at least one purple)  1  P(none picked purple )  1  (.84)(.84)(.84)  .407
The hard way:
P(at least one purple)  P(1 purple and 2 not purple) 
P(2 purple and 1 not purple)  P(3 purple)
P(at least one purple)  3(.16)(.84)(.84)  3(.16)(.16)(.84)  (.16)(.16) (.16)
P(at least one purple)  .407
Can you count the ways…..
You are purchasing a new car. The possible manufacturers, car
sizes, and colors are listed.
Manufacture: Ford, GM, Honda
Car size: compact, midsize
Color: white (W), red (R), black (B), green (G)
 How many different ways can you select one manufacturer, one
care size, and one color?
Can you count the ways…..
Let’s Diagram it!!!!!!
So the total number of ways =
Fundamental Counting Principle
If one event can occur in m ways and a second event can occur
in n ways, the number of ways the two events can occur in
sequence is m*n.
Can you count the ways…..
Using our prior example for purchasing a new car
Number of ways to choose manufacturer =
Number of ways to choose a size =
Number of ways to choose a color =
 How many different ways can you select one manufacturer, one
care size, and one color (using fundamental counting principle)?