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6.1-6.2 Probability Models

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?
the more repetition, the closer it gets to the
true proportion

- if individual outcomes are uncertain but
there is nonetheless a regular distribution of
outcomes in a large number of repetitions.
◦ 1- you must have a long series of independent
trials
◦ 2- probabilities imitate random behavior

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 longterm relative frequency.

What is a mathematical description or model
for randomness of tossing a coin?
This description has two parts.

1- A list of all possible outcomes

2- A probability for each outcome

x
P(x)
H
½
T
½




Sample space S- a list of all possible
outcomes.
Ex: S= {H,T}
S={0,1,2,3,4,5,6,7,8,9}
Event- an outcome or set of outcomes
(a subset of the sample space)
Ex: roll a 2 when tossing a number cube



If we have two dice, how many combinations can
you have?
6 * 6 = 36
If you roll a sum of five, what could the dice
read?
(1,4) (4,1) (2,3) (3,2)
How can we show possible outcomes?
list, tree diagram, table, etc….

Resembles the branches of a tree.
*allows us to not overlook things


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?
2 * 6 = 12
Ex: You flip four coins, what is your
sample space of getting a head and
what are the possible outcomes?
S= {0,1,2,3,4}
Possible outcomes: 2 * 2 * 2 * 2 = 16

0
TTTT
1
HTTT
THTT
TTHT
TTTH
2
HHTT
HTHT
HTTH
THHT
TTHH
THTH
3
THHH
HTHH
HHTH
HHHT
4
HHHH
X
0
1
2
3
4
P(x)
1/16
1/4
3/8
1/4
1/16


Ex: Generate a random decimal
number. What is the sample space?
S={all numbers between 0 and 1}
a) S= {G,F}
b) S={length of time after treatment}
c) S={A,B,C,D,F}




With replacement- same probability and the
events remain independent
Ex:
Without replacement- changes the probability
of an event occurring
Ex:
 #1)
0 ≤ P(A) ≤ 1
 #2)
P(S) = 1


#3-
#4- Disjoint- A and B have no outcomes in
common (mutually exclusive)
P(A or B)= P(A) + P(B)



Union:
“or” P(A or B) = P(A U B)
Intersect:
“and” P(A and B) = P(A ∩ B)
Empty event: (no possible outcomes)
S={ } or ∅



P(A)= 0.34
P(B)=0.25
P(A ∩ B)=0.12
Marital
Status
Never
Married
Married
Widowed
Divorced
Probability
0.353
0.574
0.002
0.071



What is the sum of these probabilities?
1
P(not married)=
1- P(M)= 1 – 0.574 = 0.426
P(never married or divorced)=
0.353 + 0.071 = 0.424
First
Digit
1
2
3
4
5
6
Prob.
.301
.176
.125
.097
.079
.067 .058



A= {first digit is 1}
P(A)=.30
B= {first digit is 6 or greater}
P(B)=.222
C={first digit is greater than 6}
P(C)=.155
7
8
9
.051 .046
D={first digit is not 1}
P(D)= 1- 0.301= 0.699
 E={1st number is 1, or 6 or greater}
P(E)=0.522
 F={ODD}
P(F)=0.609
G={odd or 6 or greater}
P(G)=0.727



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)= count of outcomes in A
count of outcomes in S







Try 6.18 with your partners
A) 0.04
B) 0.69
Try 6.19
A) 0.1
B) 0.3
C) regular: 0.5 peanut: 0.4
 Rule
5:
P(A and B)= P(A) P(B)
(only for independent events!)
6.24: One Big: 0.6
3 small: (0.8)³=0.512
6.25:
(1-0.05)^12=0.5404
6.26:
the events aren’t independent