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AMS 311
Lecture 6
February13, 2001
Today’s equiprobable model problems
Example 1: If a fair coin is tossed k times independently, what is the probability that the
first head will appear on the kth toss?
Example 2: If a man has n keys, only one of which will unlock his door, and if tries them
in a random order (without replacement), what is the probability that he will try exactly k1 wrong keys before finding the correct one?
Last Class:
Definition: If P(B)>0, the conditional probability of A given B, denoted by P(A|B), is
P( AB)
P( A | B) 
.
P( B)
Theorem 3.3. Law of Total Probability
Let B be an event with P(B)>0 and P(Bc)>0. Then for any event A,
P( A)  P( A | B) P( B)  P( A | B c ) P( B c ).
Definition: Let {B1 , B2 ,, Bn } be a set of nonempty subsets of the sample space S of an
experiment. If the events B1 , B2 ,, Bn are mutually exclusive and
n

i 1
Bi  S , the set
{B1 , B2 ,, Bn } is called a partition of S.
Theorem 3.4. Generalized Law of Total Probabililty
If {B1 , B2 ,, Bn } is a partition of the sample space of an experiment and P(Bi)>0 for
i  1,2,  , n , then for any event A of S,
n
P( A)   P ( A | Bi ) P( Bi ).
i 1
Application to overall mortality rate; definition of directly standardized rate.
Much harder example using simple law of total probability:
Example 3.14. Gambler’s Ruin Problem
Two gamblers play the game of “heads or tails,” in which each time a fair coin lands
heads up, player A wins $1 from B, and each time it lands tails up, player B wins $1 from
A. Suppose that player A initially has a dollars and player B has b dollars. If they
continue to play this game successively, what is the probability that (a) A will be ruined;
(b) the game goes forever with nobody winning?
Bayes’ Theorem
Let {B1 , B2 , , Bn } be a partition of the sample space S of an experiment. If
for i  1, 2, , n, P( Bi )  0, then for any event A of S with P(A)>0,
P( A| Bk ) P( Bk )
P( Bk | A) 
.
P( A| B1 ) P( B1 )  P( A| B2 ) P( B2 )   P( A| Bn ) P( Bn )
Example Bayes’ Theorem Problem
The probability that a randomly selected individual has a specified medical condition is
.20. A screening test for this condition has sensitivity .98. That is, the probability that an
affected individual tests positive is .98. The specificity of the procedure is .97. That is,
the probability that an individual not affected tests negative is .97 What is the probability
that the screening test will produce a false positive reading? That is, given that the results
of the test are positive, what is the probability that the individual does not have the
condition?
Definition: Two events A and B are called independent if
P( AB)  P( A) P( B).
If two events are not independent, they are called dependent. If A and B are independent,
we say that {A, B} is an independent set of events.
Reasons for events to be independent:
 Device has been constructed to have independent outcomes (roulette wheels, etc.).
 A sample has been taken following the precise rules.
 Experimental units have been randomly assigned to treatments.
To show two events are independent, apply the definition.
Example 3.31. An electric circuit has four switches that are independently closed or open
with probabilities p and 1-p respectively. If a signal is fed to the input, what is the
probability that it is transmitted to the output?