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6.1 The Idea of Probability
Objective: Understand the term “random”
Implement different probability models
Use the rules of probability in calculations
Chance behavior is unpredictable in the short run but has a regular and predictable pattern in
the long run
What does that mean to you?
Random- if individual outcomes are uncertain but there is nonetheless a regular distribution of
outcomes in a large number of repetitions.
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3Probability- The probability of any outcome of a random phenomenon is the proportion of
times the outcome would occur in a very long series of repetitions. That is, the probability is
long-term relative frequency.
6.2 Probability Models
What is a mathematical description or model for randomness of tossing a coin?
This description has two parts.
12Probability Models
Sample space SEx:
EventEx:
Ex:
If we have two dice, how many combinations can you have?
If you roll a five, what could the dice read?
How can we show possible outcomes?
Tree Diagram-
Multiplication Principle- If you can do one task in a number of ways and a second task in b
number of ways, then both tasks can be done in a x b number of ways.
Ex: How many outcomes are in a sample space if you toss a coin and roll a dice?
Ex: You flip four coins, what is your sample space of getting a head and what are the possible
outcomes?
What is the probability Distribution?
Ex: Generate a random decimal number. What is the sample space?
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6.9:
With replacementEx:
Without replacementEx:
Probability Rules
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#2-
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Union:
Intersect:
Empty event:
Venn Diagram:
Ex: Display the probabilities by using a Venn Diagram.
P(A)= 0.34
P(B)=0.25
P(A  B)=0.12
Ex:
Marital Status Never Married Married
Probability
0.353
0.574
a) What is the sum of these probabilities?
Widowed
0.002
Divorced
0.071
b) P(not married)=
c) P(never married or divorced)=
Probabilities in a finite sample space:
Assign a probability to each individual outcome. These probabilities must be numbers between
0 and 1 and must have sum 1.
The probability of any event is the sum of the probabilities of the outcomes making up the
event.
Benford’s Law
First Digit: 1
Probability 0.301
2
0.176
3
0.125
4
0.097
A= {first digit is 1}
P(A)=
B= {first digit is 6 or greater}
P(B)=
C={first digit is greater than 6}
P(C)=
D={first digit is not 1}
P(D)=
E={first number is 1 or 6 or greater}
P(E)=
F={ODD}
P(F)=
5
0.079
6
0.067
7
0.058
8
0.051
9
0.046
G={odd or 6 or greater}
P(G)=
Equally likely outcomes
If a random phenomenon has k possible outcomes, all equally likely, then each individual
outcome has probability 1/k. The probability of any event A is:
P(A)=
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6.19:
The Multiplication Rule for Independent Events
Rule 5:
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