Download Introduction to probability

1. Axioms of probability
1.2 Sample space and events
Experiment (eg. Tossing a die)
Outcome(sample point)
Sample space={all outcomes}
Event: subset of sample space
•
Ex1.1 tossing a coin once
sample space S = {H, T}
• Ex1.2 flipping a coin and tossing a die if T
or flipping a coin again if H
S={T1,T2,T3,T4,T5,T6,HT,HH}
p2.
Sample space and events
•
Ex1.3 measuring the lifetime of a light bulb
S={x: x
E={x: x
•
 0}
 100} is the event that the light
bulb lasts at least 100 hours
Ex1.4 all families with 1, 2, or 3 children
(genders specified)
S={b,g,bg,gb,bb,gg,bbb,bgb,bbg,bgg,
ggg,gbg,ggb,gbb}
p3.
Sample space and events


Event E has occurred in an experiment:
If the outcome of an experiment belongs to E.
Take events E, F as sets and sample space S then
E  F , EF  E  F , E  F , E  F , E c  S  E

can be defined straightforward.
Can also define
 E
ni1 Ei ,ni1 Ei ,
E
,

i 1 i i 1 i
if {E1, E2, … } is a set of events
p4.
Sample space and events

Associative laws EU(FUG)=(EUF)UG
Distributive laws (EF)UH=(EUH)(FUH)
(EUF)H=(EH)U(FH)

De Morgan’s 1st law: (E U F)c = EcFc

De Morgan’s 2nd law: (EF)c = Ec U Fc

E = ES = E(FUFc) = EF U EFc

p5.
1.3 Axioms of probability
Definition(Probability Axioms)
S: sample space
A: event, A  S
Pr: a function for each event A, i.e. Pr: 2S  R
Pr(A) is said to be the probability of A if
Axiom 1 Pr(A)  0
Axiom 2 Pr(S) = 1
Axiom 3 If {A1, A2, A3, … } is a sequence of
mutually exclusive events then
Pr( 
i 1 Ai ) 

Pr( Ai )
i 1

p6.
1.4 Basic Theorem




Theorem 1.4 P(Ac) = 1 – P(A)
Theorem 1.5 If A  B, then
P(B-A)=P(BAc)=P(B)-P(A)


Corollary If A B, then P(A)
P(B)
Theorem 1.6 P(AUB) = P(A)+P(B)-P(AB)
(see p.19)
p7.
Basic Theorem

Ex 1.15 In a community of 400 adults, 300 bike or
swim or do both, 160 swim, and 120 swim and
bike. What is the probability that an adult,
selected at random from this community, bike?
Sol: A: event that the person swims
B: event that the person bikes
P(AUB)=300/400, P(A)=160/400,
P(AB)=120/400
P(B)=P(AUB)+P(AB)-P(A)
= 300/400+120/400-160/400=260/400= 0.65
p8.
Basic Theorem

Ex 1.16 A number is chosen at random from the
set of numbers {1, 2, 3, …, 1000}. What is the
probability that it is divisible by 3 or 5(I.e. either
3 or 5 or both)?
Sol: A: event that the outcome is divisible by 3
B: event that the outcome is divisible by 5
P(AUB)=P(A)+P(B)-P(AB)
=333/1000+200/1000-66/1000
=467/1000
p9.
Basic Theorem

Inclusion-Exclusion Principle
P( A1  A2  ...  An ) 


 P( Ai )   P( Ai A j )
P( Ai A j Ak )  ...  ( 1)n 1 P( A1A2 ... An )
p10.
1.5 Continuity of probability
function

Recall the continuity of a function f: RR
lim f ( xn )  f ( lim xn )


n
n
fro every convergent seq {xn} in R.
The continuity of probability function is similar.
Def. A seq {En, n>=1} of event of a sample space
is called increasing if
E1  E2  E3   En  En 1  ;
it is called decreasing if
E1  E2  E3   En  En 1  .
p11.
Continuity of probability function

Thm 1.8(continuity of probability function)
For any increasing or decreasing sequence of
events, {En, n>=1}:
lim P(En)=P(lim En)
p12.
1.6 Probabilities 0 and 1


1.
2.
If E and F are events with probabilities 1 and 0,
then it is not correct to say that E is the sample
space S and F is the empty set.
Example: selecting a random point from (0,1)
A={1/3}, P(A)=0
B=(0,1)-A, P(B)=1
p13.
1.7 Random selection of points
from intervals

Def. A point is randomly selected from an interval
(a, b). The probability the subinterval (c, d)
contains the point is defined to be (d-c)/(b-a).
p14.