Transcript Probability - University of Rhode Island

Intro to Probability
STA 220 – Lecture #5
Randomness and Probability
• We call a phenomenon
if individual
outcomes are uncertain but there is
nonetheless a regular distribution of
outcomes in a large number of repetitions
• The
of any outcome of a
random phenomenon is the proportion of
times the outcome would occur in a very long
series of repetitions
Probability Models
• The description of a random phenomenon in
the language of mathematics is called a
• A probability model consists of 2 parts:
– A list of
–A
for each outcome
Probability Models
• Example: Toss a coin.
• We do not know
• But we do know:
– The outcome will be either heads or tails
– We believe that each of these outcomes has a
probability of ½
Probability Models
• The
of a random phenomenon
is the set of all possible outcomes
• Example: Toss a coin
– S = {heads, tails} or S = {H, T}
• Example: Toss a coin 4 times. Count # of Heads
– S = {0,1,2,3,4}
• Example: Roll a die
– S=
Probability Models
• Example: Suppose that in conducting an
opinion poll you select four people at random
from a large population and ask each if he or
she favors reducing federal spending on lowinterest student loans. The answers are “Yes”
or “No”. Interested in the number of “Yes”
responses.
–S=
Intuitive Probability
• An
is an outcome or a set of
outcomes of a random phenomenon. That is,
an event is a subset of the sample space.
• In a probability model, events have
Intuitive Probability
• Probability Rules
1. The probability P(A) of any event A satisfies
2. If S is the sample space in a probability model,
then P(S) =
3. Two events A and B are
if they have no outcomes
in common and so can never occur together. If A and B
are disjoint,
P(A or B) =
4. The
of any event A is the event that A does
not occur, written as AC. The complement rule states that
P(AC) = 1 – P(A)
Intuitive Probability
• A picture that shows the sample space S as a
rectangular area and events as areas within S
is called a
S
A
B
Intuitive Probability
• Venn Diagram for
events A and B
S
A
B
Intuitive Probability
• Venn Diagram for the
A
of A
Ac
Intuitive Probability
• Example
– Distance learning courses are rapidly gaining
popularity among college students. Choose at
random an undergraduate taking a distance
learning course for credit, and record the
student’s age. Here is the probability model:
Age Group
18 to 23
Years
24 to 29
Years
30 to 39
Years
40 years or
over
Probability
0.57
0.17
0.14
0.12
Intuitive Probability
Age Group
18 to 23
Years
24 to 29
Years
30 to 39
Years
40 years or
over
Probability
0.57
0.17
0.14
0.12
The probability that the student we draw is not in the
traditional undergraduate age range of 18 and 23 years
is, by the complement rule,
P(not 18 to 23 years) =
= 1 – 0.57
= 0.43
Intuitive Probability
Age Group
18 to 23
Years
24 to 29
Years
30 to 39
Years
40 years or
over
Probability
0.57
0.17
0.14
0.12
The events “30 to 39 years” and “40 years or over” are
disjoint because no student can be in both age groups.
So the addition rule says:
P(not 30 years or over) =
= 0.14 + 0.12
= 0.26
Finite Sample Space
• 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
Finite Sample Space
• Example
– Faked numbers in tax returns, payment records,
invoices, expense account claims, and many other
settings often display patterns that aren’t present
in legitimate records. Some patterns, like too
many round numbers, are obvious and easily
avoided by a clever crook. Others are more
subtle. It is a striking fact that the first digits of
numbers in legitimate records often follow a
distribution known as
Finite Sample Space
• Benford’s law
First Digit
1
2
3
4
5
6
7
8
9
Probability
0.301
0.176
0.125
0.097
0.079
0.067
0.058
0.051
0.046
Benford’s law usually applies to the first digits of the sizes
of similar quantities, such as invoices, expense account
claims, and county populations. Investigators can detect
fraud by comparing the first digits in records such as
invoices paid by a business with these probabilities.
Finite Sample Space
First Digit
1
2
3
4
5
6
7
8
9
Probability
0.301
0.176
0.125
0.097
0.079
0.067
0.058
0.051
0.046
• Consider the events
– A = (first digit is 1)
– B = (first digit is 6 or greater)
• From the table of probabilities,
– P(A) = P(1) =
– P(B) = P(6)+ P(7)+ P(8)+ P(9)
=
= 0.222
Finite Sample Space
• The probability that a first digit is anything other
than 1 is, by the complement rule,
P(Ac) = 1 – P(A)
= 1 – 0.301 =
• The events A and B are disjoint, so the probability
that a first digit is either 1 or 6 or greater is, by
the addition rule,
P(A or B) =
= 0.301 + 0.222 = 0.523
Finite Sample Space
• Be careful to apply the addition rule only to
disjoint events. Check that the probability of the
event C that a first digit is odd is
P(C) = P(1)+ P(3)+ P(5)+ P(7)+ P(9)=
• The probability
P(B or C) = P(1)+ P(3)+ P(5)+ P(6)+
P(7)+ P(8)+ P(9)=
is not the sum of the P(B) and P(C), because
events B and C are not disjoint. Outcomes and
are common to both events.
Equally likely outcomes
• In some circumstances, we are willing to
assume that individual outcomes are equally
likely because of some balance in the
phenomenon
• Examples:
– Ordinary coins have a physical balance that should
make heads and tails equally likely
– The table of random digits comes from a
deliberate randomization
Equally likely outcomes
• Example
– You might think that first digits are distributed “at
random” among the digits 1 to 9. The 9 possible
outcomes would then be equally likely.
– The sample space for a single first digit is:
S=
– Because the total probability must be 1, the
probability of each of the 9 outcomes must be
Equally likely outcomes
• The probability of the event B that a randomly
chosen first digit is 6 or greater is
P(B) = P(6) + P(7) + P(8) + P(9)
=
= 4/9 = 0.444
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
count of outcomesin A
P(A)
count of outcomesin S
P(A)