Download Statistics 510: Notes 1

Statistics 510: Notes 7
Reading: Sections 3.4, 3.5
Next week’s homework will be e-mailed and posted on the
web site www-stat-wharton.upenn.edu/~dsmall/stat510-f05
by tonight.
I. Independent Events
Review:
E is independent of F if knowledge that F has occurred
does not change the probability that E occurs, i.e.,
P( E | F )  P( E )
Independence can be expressed in the following way:
Two events E and F are said to be independent if
P( E  F )  P( E ) P( F ) .
Two events E and F that are not independent are said to be
dependent.
Example 1: Suppose that we toss two fair dice (green and
red). Let E be the event that the sum of the two dice is 6
and F be the event that the green die equals 4. Are E and
F independent?
Suppose E is independent of F. We will now show that E
is also independent of F C .
Proof: Assume that E is independent of F. Since
E  ( E  F )  ( E  F C ) , and E  F and E  F C are
mutually exclusive, we have that
P( E )  P( E  F )  P( E  F C )
 P( E ) P( F )  P( E  F C )
or equivalently,
P( E  F C )  P( E )[1  P( F )]
 P( E ) P( F C )
By similar reasoning, it follows that if E is independent of
F, then (i) E C is independent of F and (ii) E C is
independent of F C .
Independence for more than two events
Independence becomes more complicated when we
consider more than two events. We consider events
E1 , , En to be mutually independent if knowing that some
subset of the events Ei1 , , Eir has occurred does not affect
the probability that an event E j has occurred where
j  i1 , , ir .
Consider three events E , F , G . Does pairwise
independence of the events (i.e., E is independent of F ,
E is independent of G and F is independent of G )
guarantee mutual independence? No.
Example 2: A fair coin is tossed twice. Let E denote the
event of heads on the first toss, F denote the event of heads
on the second toss, and G denote the event that exactly one
head is thrown. Verify that E , F , G are pairwise
independent but that P(G | E  F )  P(G ) .
We define events E1 , , En to be mutually independent if
P( Ei1  Ei2   Eir )  P( Ei1 ) P( Ei2 ) P( Eir )
for every subset Ei1 , , Eir of these events. If E1 , , En are
mutually independent , then knowing that some subset of
the events Ei1 , , Eir has occurred does not affect the
probability that an event E j has occurred where
j  i1 ,
, ir .
Example 3: Suppose that a fair coin is flipped three times.
Let H1 be the event of a head on the first flip; T2 a tail on
the second flip; and H 3 a head on the third flip. Are H1 ,
T2 and H 3 mutually independent?
Example 4: Recall the Los Angeles Times (August 24,
1987) article from Notes 3 on the infectivity of AIDS
“Several studies of sexual partners of people infected
with the virus show that a single act of unprotected vaginal
intercourse has a surprisingly low risk of infecting the
uninfected partner – perhaps one in 100 to one in 1000.
For an average, consider the risk to be one in 500. If there
are 100 acts of intercourse with an infected partner, the
odds of infection increase to one in five.
Statistically, 500 acts of intercourse with one infected
partner or 100 acts with five partners lead to a 100%
probability of infection (statistically, not necessarily in
reality).”
Suppose that virus transmissions in 500 acts of intercourse
are mutually independent events and that the probability of
transmission in any one act is 1/500. Under this model,
what is the probability of infection?
Repeated Independent Trials
Example 4 is a special case of a common setup in which the
overall probability experiment consists of a sequence of
identical, independent subexperiments (In Example 3 the
subexperiments are whether virus is transmitted in one act
of intercourse).
Example 5: On her way to work, a commuter encounters
four traffic signals. The distance between each of the four
is sufficiently great that the probability of getting a green
light at any intersection is independent of what happened at
any prior intersection. If each light is green for 40 seconds
of every minute, what is the probability that the driver has
to stop at least three times?
Repeated independent trials problems sometimes involve
experiments consisting of a countably infinite number of
subexperiments. Solving these problems often requires
using the formula for the sum of a geometric series:
if 0  p  1,

1
k
p


1 p
k 0
Example 6: Independent trials, consisting of rolling a pair
of fair dice are performed. What is the probability that an
outcome of 5 appears before an outcome of 7 when the
outcome of a roll is the sum of the dice?
II. P ( | F ) is a probability
The conditional probability P ( | F ) is a probability function
on the events in the sample space S and satisfies the usual
axioms of probability:
(a) 0  P( E | F )  1
(b) P ( S | F )  1
(c) If Ei , i  1, 2, are mutually exclusive events, then

P(
1

Ei | f )   P( Ei | F )
1
Thus, all the formulas we have derived for manipulating
probabilities in Chapter 2 apply to conditional probabilities.
Example 7: The following is a simple genetic model.
Assume that genes in an organism occur in pairs and that
each member of the pair can be either of the types a or A.
The possible genotypes of an organism are then AA, Aa and
aa (Aa and aA are equivalent). When two organisms mate,
each independently contributes one of its two genes; either
one of the pair is transmitted with probability .5.
A female chimp has given birth. It is not certain, however,
which of two male chimps is the father. Before any genetic
analysis has been performed, it is felt that the probability
that male number 1 is the father is p, and the probability
that male number 2 is the father is 1-p. DNA obtained
from the mother, male number 1 and male number 2
indicate that on one specific location of the genome the
mother has the gene pair AA, male number 1 has the gene
pair aa and male number 2 has the gene pair Aa. If it
results that the baby chimp has the gene pair Aa, what is the
probability that male number 1 is the father?
Conditional independence: An important concept in
probability theory is that of the conditional independence of
events. We say that events E1 and E2 are conditionally
independent given F if, given that F occurs, the conditional
probability that E1 occurs is unchanged by information as
to whether or not E2 occurs.
More formally, E1 and E2 are said to be conditionally
independent given F if
P( E1 | E2  F )  P( E1 | F )
or, equivalently,
P( E1  E2 | F )  P( E1 | F ) P( E2 | F ) .
Example 8: An insurance company believes that people can
be divided into two classes: those who are accident-prone
and those who are not. Their statistics show that an
accident-prone person will have an accident at some time
within a fixed 1-year period with probability .4, whereas
this probability decreases to .2 for a non-accident prone
person. 30 percent of the population is accident-prone.
Consider a two-year period. Assume that the event that a
person has an accident in the first year is conditionally
independent of the event that a person has an accident in
the second year given whether or not the person is accident
prone. What is the conditional probability that a randomly
selected person will have an accident in the second given
that the person had an accident in the first year?