Download Notes 7 - Wharton Statistics

Statistics 510: Notes 7
Reading: Section 3.5.
I. P ( | F ) is a probability and Conditional Independence.
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.
We can also define conditional independence. 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,
1
P( E1  E2 | F )  P( E1 | F ) P( E2 | F ) .
Example 1: 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 year
given that the person had an accident in the first year?
2
II. Recursive Formulas for Computing Probabilities
A recursive formula is a formula that is used to determine
the next term of a sequence using one or more of the
preceding terms, e.g., a recursive formula for the sequence
5, 20, 80, 320 is xn  4 xn1
Recursive formulas are often useful for computing
probabilities, especially in connections with conditional
probability.
Example 2: Success runs. A basketball player has a 50%
success rate in free throw shots. Assuming that the
outcomes of all free throws are independent from one
another, what is the probability that, within a sequence of
20 shots, the player makes at least five shots in a row at
some point?
3
4
Example 3 (Example 3.5d, pg. 106 in Ross): The matching
problem. At a party, n men take off their hats. The hats
are then mixed up, and each man randomly selects one.
We say that a match occurs if a man selects his own hat.
What is the probability of
(a) no matches
(b) exactly k matches
5
6
Example 3: Success runs. A basketball player has a 50%
success rate in free throw shots. Assuming that the
outcomes of all free throws are independent from one
another, what is the probability that, within a sequence of
20 shots, the player makes at least five shots in a row at
some point?
7