Mathematics, Statistics, Probability, Randomness, General Probability Rules, General Addition Rules, Conditional Probability, General Multiplication Rules, Bayes’s Rule, Independence

1. INTRODUCTION TO
STATISTICS & PROBABILITY
Chapter 4:
Probability: The Study of Randomness
(Part 4)
Dr. Nahid Sultana
1

2. Chapter 4
Probability: The Study of Randomness
4.1 Randomness
4.2 Probability Models
4.3 Random Variables
4.4 Means and Variances of Random Variables
4.5 General Probability Rules*
2

3. 4.5 General Probability Rules
3
 Rules of Probability
 General Addition Rules
 Conditional Probability
 General Multiplication Rules
 Bayes’s Rule
 Independence

4. 4
Probability Rules
Return to the laws of probabilities:
Rule 1. The probability P(A) of any event A satisfies 0 ≤ P(A) ≤ 1.
Rule 2. If S is the sample space in a probability model, then P(S) = 1.
Rule 3. If A and B are disjoint, P(A or B) = P(A) + P(B).
Rule 4. For any event A, P(AC) = 1 – P(A).
Rule 5. If A and B are independent, P(A and B) = P(A)P(B).

5. 5
The General Addition Rule
Addition Rule for Unions of Two
Events
The probability that A occurs, B occurs,
or both occurs is:
P(A or B) = P(A) + P(B) – P(A and B)
Addition Rule for Disjoint Events: If A, B, and C are disjoint in the
sense that no two have any in common, then:
P( A or B or C) = P(A) + P(B) + P(C)

6. 6
Conditional Probability
Rule 5. If A and B are independent, P(A and B) = P(A)P(B).
The probability an event can change if we know that some other event
has occurred. This idea is the key to many applications of probability.
The conditional probability reflects how the probability of an event
can change if we know that some other event has already occurred.
When P(A) > 0, the conditional probability of B given A is
)(
)and(
)|(
AP
BAP
ABP =

7. 7
The General Multiplication Rule
The probability that any two events A and B both occur is
P(A and B) = P(A) P(B | A)
This is the general multiplication rule, where P(B | A) is the conditional
probability that event B occurs given that event A has already occurred.
Conditional probability  rule for the probability that both of two event
occur.
Note: If Two events A and B are independent, P(A and B) = P(A)P(B).
Therefore, If A and B that both have positive probability are
independent if:
P(B|A) = P(B)
)(
)and(
)|(
AP
BAP
ABP =

8. 8
The General Multiplication Rule
Example:
Suppose 29% of Internet users download music files, and 67% of
downloaders say they don’t care if the music is copyrighted.
What is the percent of Internet users who download music and don’t care
about copyright.
Example:
Employed Unemployed Total
Male 460 40 500
Female 140 260 400
Total 600 300 900

9. 9
Probability Trees
Conditional probabilities can get complex, and it is often good strategy
to build a probability tree that represents all possible outcomes
graphically and assigns conditional probabilities to subsets of events..
Consider flipping a coin
twice.
What is the probability of
getting two heads?
Sample Space:
HH HT TH TT
So, P(two heads) = P(HH) = 1/4

10. 10
Probability Trees: Example
The Pew Internet and American Life Project finds that 93% of
teenagers (ages 12 to 17) use the Internet, and that 55% of online
teens have posted a profile on a social-networking site.
What percent of teens are online and have posted a profile?
55.0)online|profile(
93.0)online(
=
=
P
P
51.15% of teens are online and have
posted a profile.
P(A and B) = P(A) P(B | A)
5115.0
)55.0)(93.0(
online)|(profile(online)
profile)haveand(online
=
=
⋅= PP
P

11. 11
Probability Trees: Example
29% of adult Internet users are 18 to 29 years old (event A1), another 47%
are 30 to 49 (event A2), and the remaining 24% are 50 and over (event A3).
47% of the 18 to 29 age group chat, as do 21% of the 30 to 49 age
group and just 7% of those 50 and over. What is the probability that a
randomly chosen user of the Internet participates in chat rooms?
P(C) = 0.1363 + 0.0987 + 0.0168
= 0.2518

12. 12
Bayes’s Rule
If a woman in her 20s gets screened for breast cancer and receives a
positive test result, what is the probability that she does have breast
cancer?
Cancer
No cancer
Mammography
Positive
Negative
Positive
Negative
Disease
incidence
Diagnosis
sensitivity
Diagnosis
specificity
False negative
False positive
0.0004
0.9996
0.8
0.2
0.1
0.9
Incidence of breast cancer
among women ages 20–30
Mammography
performance
Example

13. 13
Bayes’s Rule (Cont…)
Example
If a woman in her 20s gets screened for breast cancer and receives a positive test
result, what is the probability that she does have breast cancer?
A1 is cancer, A2 is no cancer, C is a positive test result.
%3.0
)9996.0(1.0)004.0(8.0
)0004.0(8.0
)cancerno()cancerno|pos()cancer()cancer|pos(
)cancer()cancer|pos(
)pos|cancer(
≈
+
=
+
=
PPPP
PP
P
Cancer
No cancer
Mammography
Positive
Negative
Positive
Negative
Disease
incidence
Diagnosis
sensitivity
Diagnosis
specificity
False negative
False positive
0.0004
0.9996
0.8
0.2
0.1
0.9
Incidence of breast
cancer among
women ages 20–30 Mammography
performance

14. 14
Bayes’s Rule (Cont…)
 If a sample space is decomposed in k disjoint events, A1, A2, … , Ak
none with a null probability and P(A1) + P(A2) + … + P(Ak) = 1
 And if C is any other event such that P(C) is not 0 or 1, then:
However, it is often intuitively much easier to work out answers with a
probability tree than with these lengthy formulas.