The law of total probability
Polya’s urn scheme
Baye’s formula
§1.4 The Total Probability Theorem
and
Baye’s Rule
Tom Lewis
Fall Semester
2016
The law of total probability
Polya’s urn scheme
Outline
The law of total probability
Polya’s urn scheme
Baye’s formula
Baye’s formula
The law of total probability
Polya’s urn scheme
Baye’s formula
Theorem (Law of total probability)
Let A1 , A2 , . . . , An be a partition of Ω and let B be an event.
Then
n
X
P (B ) =
P (B | Ak )P (Ak ).
k =1
The law of total probability
Polya’s urn scheme
Baye’s formula
Problem
On any given day it will rain with probability .3 and shine with
probability .7. If it rains, then I will be late with probability .8. If it
shines, then I will be late with probability .1. What is the
probability that I will be late on any given day?
The law of total probability
Polya’s urn scheme
Baye’s formula
Problem
An urn contains r + b balls: r red and b black. An experiment
consists of selecting a ball from the urn, noting its color and then
replacing this ball along with c of the same color to urn. Suppose
that this experiment is repeated n times and let
Ri = a red ball is drawn on the i th try
Bi = a black ball is drawn on the i th try
Find P (R1 ), P (B1 ), P (R2 ), P (B2 ).
The law of total probability
Polya’s urn scheme
Baye’s formula
Problem
Alice, Bob, and Charlie play a game; they take turns tossing a fair
coin; the winner is the first person to toss a head. If they go in the
order Alice, Bob, and Charlie, then what are their probabilities of
winning the game?
The law of total probability
Polya’s urn scheme
Baye’s formula
Theorem (Baye’s Theorem)
Let A1 , A2 , . . . , An be a partition of Ω and let B be an event.
Then
P (B | A1 )P (A1 )
P (A1 | B ) = Pn
.
k =1 P (B | Ak )P (Ak )
The law of total probability
Polya’s urn scheme
Baye’s formula
The meaning of Baye’s Theorem
Think of the events A1 , · · · , An as a complete set of “causes” of
the event B ; thus, suppose we know
P (B | A1 ),
P (B | A2 ),
...
, P (B | An ).
In a Baye’s calculation, the order of causality is twisted; the formula
gives us the recipe for untwisting the events.
The law of total probability
Polya’s urn scheme
Baye’s formula
Problem (Baye’s formula)
Urn 1 contains five balls: 4 white, 1 black. Urn 2 contains six balls:
2 white, 4 black. An experiment consists of selecting an urn at
random and then selecting a ball from this urn. Given that you
have selected a black ball, what is the probability that it came from
urn 2?
The law of total probability
Polya’s urn scheme
Baye’s formula
Problem (False positive paradox)
• Let us suppose that 2% of the population has used marijuana
in the last week.
• Let us suppose that the K-9 patrol have a marijuana sniffing
dog, Fido, who is trained to growl when he detects the
presence of marijuana.
• Fido is not perfect. If someone has used marijuana in the last
week, then 90% of the time Fido will growl. If someone has
not used marijuana in the last week, then 8 % of the time Fido
will still growl.
• If a random citizen is presented to Fido and he growls, then
what is the chance he has actually used marijuana in the last
week?