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AMS 311
Lecture
February10, 2000
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)
Let’s Make a Deal Example.
This Class:
Extra Credit Project Suggestion:
The New York Times, January 25, 2000 (page E5), had a story reporting on a concept
called “Parrando’s Paradox.” The claim is “that two games guaranteed to make a player
lose all his money can generate a winning streak if played alternately.” The story reports
Dr. Sergei Maslov, a physicist at Brookhaven National Laboratory, showed “that if an
investor simultaneously shared capital between two losing stock portfolios, capital would
increase rather than decrease.”
What is Parrando’s Paradox?
Does it actually say this?
Is this true?
Does it have applications?
Law of Multiplication: P( AB)  P( B) P( A | B).
Remember that we used this in the Let’s Make a Deal example.
Theorem 3.2. (Generalization of the Law of Multiplication):
If P( A1 A2  An1 )  0, then
P( A1 A2 A3  An1 An )  P( A1 ) P( A2 | A1 ) P( A3 | A2 A1 ) P( An | A1 A2  An1 ).
Example 3.11. Suppose that five good and two defective fuses have been mixed up. To
find the defective ones, we test them one-by-one, at random and without replacement.
What is the probability that we find both of the defective fuses in exactly three tests?
0.095.
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 ).
We used this fact in solving the Let’s make a deal example.
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 dollears 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?