Transcript Example

PROBABILITY AND
STATISTICS
WEEK 2
Onur Doğan
Introduction to Probability
• The Classical Interpretation of Probability
• The Frequency Interpretation of Probability
• The Subjective Interpretation of Probability
Onur Doğan
Sample Space, Experiment, Event
An experiment is any process, real or hypothetical,
in which the possible outcomes can be identified
ahead of time.
An event is a well-defined set of possible outcomes
of the experiment.
Onur Doğan
The Sample Space of an Experiment
Onur Doğan
Recall: Operations of Set Theory
Example
 Example: Consider tossing a fair coin. Define the event H
as the occurrence of a head. What is the probability
of the event H, P(H)?
1. In a single toss of the coin, there are two possible outcomes
2. Since the coin is fair, each outcome (side) should have an equally likely chance of
occurring
3. Intuitively, P(H) = 1/2 (the expected relative frequency)
Notes:

This does not mean exactly one head will occur in every two tosses of the coin

In the long run, the proportion of times that a head will occur is approximately 1/2
Experiment
• Experimental results of tossing a coin 10 times each trial:
Trial
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Number of
Relative
Cumulative
Heads Observed Frequency Relative Frequency
5
5/10
5/10 = 0.5000
4
4/10
9/20 = 0.4500
4
4/10
13/30 = 0.4333
5
5/10
18/40 = 0.4500
6
6/10
24/50 = 0.4800
7
7/10
28/60 = 0.4667
6
6/10
34/70 = 0.4857
4
4/10
38/80 = 0.4750
7
7/10
45/90 = 0.5000
3
3/10
48/100 = 0.4800
4
4/10
52/110 = 0.4727
6
6/10
58/120 = 0.4838
7
7/10
65/130 = 0.5000
4
4/10
69/140 = 0.4929
3
3/10
72/150 = 0.4800
7
7/10
79/160 = 0.4938
6
6/10
85/170 = 0.5000
3
3/10
88/180 = 0.4889
6
6/10
94/190 = 0.4947
4
4/10
98/200 = 0.4900
Probabilities
0,8
0,7
0,6
0,5
0,4
0,3
0,2
Expected value = 1/2
0,1
0
0
5
10
15
Trial
20
25
Axioms and Basic Theorems
Onur Doğan
mutually exclusive / exhaustive
Onur Doğan
Example
Two dice are cast at the same time in an experiment.
• Define the sample space of the experiment.
• Find the pairs whose sum is 5 (A) and the pairs whose
first die is odd (B).
• Are A and B mutually exclusive?
• Are A and B exhaustive?
Onur Doğan
Example
In a city, 60% of all households subscribe to the
newspaper A, 80% subscribe newspaper B, and 10% of
all households do not subscribe any newspaper.
If a household is selected at random,
• What is the probability that it subscribes to at least one of
the two newspapers?
• Exactly one of the two newspapers?
Onur Doğan
Example
a. What is the probability that the individual has a medium auto deductible and a
high homeowner’s deductible?
b. What is the probability that the individual has a low auto deductible? A low
homeowner’s deductible?
c. What is the probability that the individual is in the same category for both
auto and homeowner’s deductibles?
d. What is the probability that the two categories are different?
e. What is the probability that the individual has at least one low deductible
level?
f. What is the probability that neither deductible level is low?
Onur Doğan
Counting Techniques
Onur Doğan
Example
Suppose that a club consists of 25 members
and that a president and a secretary are to be
chosen from the membership. We shall
determine the total possible number of ways in
which these two positions can be filled.
Onur Doğan
Permutations
Suppose that four-letter words of lower case alphabetic
characters are generated randomly with equally likely
outcomes. (Assume that letters may appear repeatedly.)
•How many four-letter words are there in the sample space S?
•How many four-letter words are there are there in S that start
with the letter ”k”?
•What is the probability of generating a four-letter word that
starts with an ”k” ?
Onur Doğan
Permutations
How many words of length 4 can be formed from a set of n
(distinct) characters,, when letters can be used at most once ?
How many words of length k can be formed from a set of n
(distinct) characters, (where k ≤ n), when letters can be used at
most once ?
Onur Doğan
Permutations
An ordered subset is called a permutation. The
number of permutations of size k that can be
formed from the n individuals or objects in a
group will be denoted by P(n,k).
Onur Doğan
Permutations
•Sampling with Replacement.
•Obtaining Different Numbers.
•Birthday Problem?
Onur Doğan
Combination
Consider a set with n elements. Each subset of
size k chosen from this set is called a
combination of n elements taken k at a time.
Onur Doğan
Example
Suppose that a club consists of 25 members
and that a president and a secretary are to be
chosen from the membership. We shall
determine the total possible number of ways in
which two people will fill the two positions.
Onur Doğan
Example
Suppose that a class contains 15 boys and 30
girls, and that 10 students are to be selected at
random for a special assignment. We shall
determine the probability exactly three boys
will be selected.
Onur Doğan
Example
Suppose that a deck of 52 cards containing
four aces is shuffled thoroughly and the cards
are then distributed among four players so that
each player receives 13 cards. We shall
determine the probability that each player will
receive one ace.
Onur Doğan
Example
Suppose that a fair coin is to be tossed 7 times,
and it is desired to determine;
(a) the probability p of obtaining exactly
three heads
(b) the probability p of obtaining three or
fewer heads.
Onur Doğan
Example
• A box containing 3 red, 4 blue and 5 green
balls. What’s the probability that drawn 3 balls
will be different colours?
Onur Doğan
Binomial Coefficients
Multinomial Coefficients
Multinomial Coefficients
Example
Example
How many nonnegative integer solutions are
there to
x + y + z = 13 ?
Onur Doğan
Example
• What is the probability the sum is 9 in three
rolls of a die ?
Onur Doğan