e fruit E Events E re E PREE i PREE The complement of event 1 - - PDF document
Conditional probability July 20g 2020 The Monty Hall 1 problem 2 Conditional probability 3 Bayes Rude Probability Rule 4 Total time Last Sf Sample space Sample Point Outcome Wj w we was 1 1 g n WA of Prc wid f l wn n E Pr 1 wi is I
Conditional probability
1
The Monty Hall
problem
2 Conditional probability
3 Bayes Rude
4 Total
Probability Rule
Last
time
Wj
Sample Point Outcome
we w was
1
1
g n
WA
wn
n
E Pr
wi
1
is I
Events E
efruit
The complement of event
E re E
PREE
i PREE
1 The
Monty Hall Problem
I
z
z
There
are
three doors
A
car is behind one of
as a
the doors
Goats behind the
two doors
1 The contestant picks
a door
but does not open it
2 Then
Hall'sassistant opens
doors revealing agoat
3
The contestant is then given the
sticking with their current choice
the other unopened door
4 He she win
the car
if andonly if
their
chosen
door is the
correct
Question
Does
the
contestant have a better
chance of winning
if
he she switches door
Assume the
Prize
is equally likely to
be
behind any
space
I 2
3
a
wz
cg
ay
g g
c
W LO G
we
assume
w
happens
I 2
3
e
E i winning by switching
3
A
2
I
3
2
I
pV ED Pr
I 2
Pr
3
O
t
15 23
Pr
l
Pr
I
13
1 13
Pr
3
i
x
is
not
switching
Stick to
3 2
2
3 I
Pr
Ez
Pr
l
Pr
3
3
e Ot D
Pr
ED
P8
l
pr
2 2Jstgx0
O
pr
3
3
0 SO
we
a better chance of winning if
we
use
2 Conditionalprobability
Examples
Two
coin flips First flip is heads
Probability of two heads
R
AH HT TH TT
g uniform probability space
is
heads
A f HHy AT
r
New sample space
TH
Htt
Event B
two heads f HH
TT HT The probability of two heads if the first
flip is heads
The Probability of B given A
is PREBIA
Example
Two coin flips Atleast one of the flips is
heads
probability
two heads
HH
Event Bsf htt
The Probability of
B
Pr
BIA
A
non uniform enample
r
Prew
Red
Bene
Green
4110
Expermint
r
Green orange
t
B
A A if
3 reds
4 greens
B f3reds
N
A p
g Pz
B with PrEnJspn
a
Pa
Asfb2g3
et
joint
p µfMt
probability
p
pm
hB
Assume
WE
A
then
Pr WAA
r
PREBIA
s
PRCA
Prca
Definition
The conditional Probability of 13
given A
is pr BIA
Pr BRA
R
AAB
A given
B
Pratt
PRE
suppose I toss
3 balls into 3 bins
at the time
A
1st bin is empty
B
2ndbin is empty
what
is Pr AIB
000
z
se
r
Mls 33 27
i
2
3
A f 2,333
1131
8 AAB
1333 IA Is 23 s 8
I
1
At B
s
PKA.ms
s z8ygPrCAhB
s ftp.B
Bayesian Inference
8
A way update
knowledge after making
an
we
may have
an estimate of probability of
a
event A
prior
knowledge After
event
B
we
can update this
estimate
to PRCA1B
In this
PEAT
Prior probability
Pr AIB Posterior Probability
Example There
is
a
new
test
for
a certain disease
1 when the test applied
to
an
the
test
comes up positive
90 l
cases
101
2 when applied to
a healthy Person the
test
a
comes up negative 80
Suppose
that only
5
Population
has
this
disease
prior
A
a
random Person is tested positive what
is
the probability that
the Person
the
Let's define events
Bo
Test
is Positive
0.05
0.2
PREBIAJPREAT PVCB
Pr BI AT PREAAB
fPrCAJprEB7
Pr
I PRAT
3 Bayes Rule
B
Prc Anoypret
4 Total Probability Rule
Imagine two
bins containing
some
number of
red and green
Bini
Bin 2
One bin is chosen withequal
Bad Probability
BBB B
Bo
Then
a ball is drawn uniformly at rqndom
what is the probability that
Ya
We Picked Bin i given that
BOB
a green baltwas drain
Bz
PrEgreenrigs Pr
green ABD ePr
greenMBB
pr
Bz
Event
n
events
An
if
B
2
forall
A
Az
sn
An i An
i.e A
mutuallyexclusive
Then
s
PRIBLADPREAD
So
the bays rule
PREBIAITPREAD
g
PREBIAJJPREAD