Download §1.3 Conditional Probability

Conditional probability
Conditional probability as a measure
The multiplication principle
§1.3 Conditional Probability
Tom Lewis
Fall Semester
2016
Conditional probability
Conditional probability as a measure
Outline
Conditional probability
Conditional probability as a measure
The multiplication principle
The multiplication principle
Conditional probability
Conditional probability as a measure
The multiplication principle
Problem
An experiment consists of tossing a fair coin twice, recording the
outcomes each time. Given that one of the coins is a head, what is
the probability that the other coin is also a head?
Conditional probability
Conditional probability as a measure
The multiplication principle
Definition
Conditional probability Let A and B be events on a common
probability space, P (B ) 6= 0. The (conditional) probability of A
given that B , denoted by P (A | B ), is given by
P (A | B ) =
P (A ∩ B )
.
P (B )
Conditional probability
Conditional probability as a measure
The multiplication principle
Problem
• Three identical storage boxes are placed before you. Each box
has two drawers.
• Each drawer of one box contains a silver coin, each drawer of a
second contains a gold coin, and the remaining box contains a
gold coin in one drawer and a silver coin in the other.
• An experiment consists of selecting a box at random and then
selecting one of its drawers at random.
• Given that the selected drawer contains a gold coin, what is
the probability that the other drawer also contains a gold coin?
Conditional probability
Conditional probability as a measure
The multiplication principle
Theorem
Let (Ω, E, P ) be a probability space and let B ∈ E be an event
with P (B ) > 0. Then the set function
A 7→ P (A | B )
is a probability measure on Ω. This measure is concentrated on B
in the sense that if A ∩ B = ∅, then P (A | B ) = 0.
Conditional probability
Conditional probability as a measure
The multiplication principle
Problem
Toss a fair coin until a head appears or until the coin has been
tossed three times.
• Develop the natural probability space (Ω, E, P ) for this
experiment.
• Let B be the event that at least one tail is tossed. Develop
the probability measure P (· | B ).
Conditional probability
Conditional probability as a measure
The multiplication principle
Theorem (The multiplication principle)
Let A and B be events on a common probability space with
P (B ) > 0. Then
P (A ∩ B ) = P (A | B )P (B ).
Conditional probability
Conditional probability as a measure
The multiplication principle
Problem
In Greenville it is either sunny or overcast. My mood is either
happy or morose. Records show that it is overcast with chance .7
and sunny with chance .3. If it is overcast, then I am happy with
chance .3 and morose otherwise. If it is sunny, then I am happy
with chance .8 and morose otherwise. What is the chance that on a
randomly selected day it will be sunny and I will be morose?
Conditional probability
Conditional probability as a measure
The multiplication principle
Problem (Assessing blame)
A market buys peaches from three different orchards: A, B , and
C . 50% of the peaches come from A, 35% from B , and the rest
from C . It is known that 3% of the peaches from A are defective,
2% of the peaches from B are defective, and 8% of the peaches
from C are defective. If a peach is selected at random and found
to be defective, what is the chance that it came from orchard C ?
Conditional probability
Conditional probability as a measure
The multiplication principle
Theorem (The extended multiplication rule)
Let A1 , A2 , A3 , A4 be non-trivial events on a common probability
space.
P (A1 ∩ A2 ∩ A3 ∩ A4 )
= P (A4 | A1 ∩ A2 ∩ A3 )P (A3 | A1 ∩ A2 )P (A2 | A1 )P (A1 ).
Conditional probability
Conditional probability as a measure
The multiplication principle
Problem
An urn contains 10 balls: 7 red and 3 green. If three balls are
selected from the urn in succession and without replacement, what
is the chance that balls came out in the order red, red, green?