Team Problem
Problem 1
Albert R Meyer, May 4, 2009
lec 13M.1
Mathematics for Computer Science
MIT 6.042J/18.062J
Conditional Probability
& Independence
Albert R Meyer, May 4, 2009
lec 13M.2
Law of Total Probability
B2A
B1
B2
Albert R Meyer, May 4, 2009
A
B3
lec 13M.3
Law of Total Probability
A = (B1A)(B2A)(B3A)
Pr{A} = Pr{B1A} +
Pr{B2A} +
Pr{B3A}
Albert R Meyer, May 4, 2009
lec 13M.4
Conditional Probability: A Fair Die
Pr{roll 1} 
{1}
{1,2,3, 4,5,6}
1

6
“knowledge” changes probabilities:
Pr{roll 1 knowing rolled odd}

{1}
{1,3,5}
1

3
Albert R Meyer, May 4, 2009
lec 13M.5
Conditional Probability
Pr{A|B} is the probability
of event A, given that
event B has occurred:
Pr{A  B}
Pr{A | B} ::
Pr{B}
Albert R Meyer, May 4, 2009
lec 13M.6
Conditional Probability: A Fair Die

O: {1,3,5}
E: {2,4,6}
A: {1}
{1}

A: {2,3,4,5,6}
{3,5}
{2,4,6}
Pr{A | O} = Pr{1}/Pr{1,3,5}
Albert R Meyer, May 4, 2009
lec 13M.7
Conditional Probability: A Fair Die

O: {1,3,5}
E: {2,4,6}
A: {1}
{1}

A: {2,3,4,5,6}
{3,5}
{2,4,6}
Pr{A | O} = (1/6)/(1/2) = 1/3
Pr{A | O} = (1/3)/(1/2) = 2/3
Pr{A | E} = 0/(1/2)
=0
Pr{A | E} = (1/2)/(1/2)
=1
Albert R Meyer, May 4, 2009
lec 13M.8
Conditional Probability: A Fair Die
Pr{one | odd)} = 1/3 Yes
Yes
1/2
{1,2,3,4,5,6}
1/2
{1}
Pr:
1/6
{3,5}
1/3
{2,4,6}
1/2
{1,3,5}
No
2/3
Pr{not one | odd} =
No
{2,4,6}
Pr{not one | even} =
Rolled odd
No
1
Rolled 1
Albert R Meyer, May 4, 2009
lec 13M.9
Product Rule
Pr{A  B} 
Pr{A| B}  Pr{B}
Albert R Meyer, May 4, 2009
lec 13M.10
Law of Total Probability
A = (B1A)(B2A)(B3A)
Pr{A} = Pr{B1A}+Pr{B2A}
+ Pr{B3A}
= Pr{A|B1 }Pr{B1} +
Pr{A|B2}Pr{B2} +
Pr{A|B3}Pr{B3}
Albert R Meyer, May 4, 2009
lec 13M.11
Conditional Probability: Monty Hall
Pr{ prize at 1 | Goat at 2} = 1/2
Really!
Outcomes:
(Prize Door, Picked Door, Carol door)
[Goat at 2] =
{ (1,1,2),(1,1,3), (1,2,3),(1,3,2)
(3,3,1),(3,3,2),(3,1,2),(3,2,1)}
Albert R Meyer, May 4, 2009
lec 13M.14
Conditional Probability: Monty Hall
Pr{ prize at 1 | Carol opens 2} = 1/2
Outcomes:
(Prize Door, Picked Door, Carol door)
[Carol opens 2] =
{ (1,1,2),
(1,1,2), (1,3,2)
(1,3,2) ,
(3,3,2),(3,1,2)}
Albert R Meyer, May 4, 2009
lec 13M.16
Conditional Probability: Monty Hall
Seems that the contestant may
as well stick, since the probability
is 1/2 given what he knows when
he chooses.
But wait, contestant knows more
than door opened by Carol: he knows
which door he picked!
Albert R Meyer, May 4, 2009
lec 13M.18
Conditional Probability: Monty Hall
Pr{ prize at 1 | picked 1 &
Carol opens 2} = 1/3
[picked 1 & Carol opens 2] =
{ (1,1,2)
(1,1,2),(3,1,2) }
Pr=1/18 Pr=1/9
1 / 18

1 / 18  1 / 9
Albert R Meyer, May 4, 2009
lec 13M.20
Independence
Definition 1:
Events A and B are independent iff
Pr{A} = Pr{A | B}.
Definition 2:
Events A and B are independent iff
Pr{A}  Pr{B} = Pr{A B}.
Albert R Meyer, May 4, 2009
lec 13M.21
Definitions of Independence
need Pr{B} ≠ 0 for Def. 1.
Def. 2 always works:
Pr{A}Pr{B} = Pr{A B}
Albert R Meyer, May 4, 2009
lec 13M.23
Independence
Pr{A}Pr{B} = Pr{A B}
symmetric in A and B so,
A independent of B iff
B independent of A.
Albert R Meyer, May 4, 2009
lec 13M.24
Independent Events?
B: Baby born at Mass General Hospital
between 1:00AM and 1:01AM.
F: Jupiter’s moon IO is full.
Albert R Meyer, May 4, 2009
lec 13M.30
Independent Events?
Does event B (baby born)
have anything to do with
event F (IO is full)?
Albert R Meyer, May 4, 2009
lec 13M.31
Babies & Full Moons
My sweet Aunt Daisy believed in
Astrology. She thought celestial
events could influence babies.
We might say “nonsense,” there’s
no effect.
But there is an effect:
Albert R Meyer, May 4, 2009
lec 13M.34
C:\42\pub\jup-radio_070115.htm
** INFORMATION FOR AMATEUR
RADIO ASTRONOMERS ** JUPITER
DECAMETRIC EMISSIONS **
JUPITER EPHEMERIS 01 Jul 1994,
0000UTC, Julian Day: 2449534.5, GMT
Sidereal Time: 18h35m17s ….
Albert R Meyer, May 4, 2009
lec 13M.36
C:\42\pub\jup-radio_070115.htm
SUMMARY: Jupiter's HF emissions are
…heard on earth when Jupiter's magnetic
field "sweeps" the earth every 9h55m27s
and at other times when Io's geometric
position influences activity.
Albert R Meyer, May 4, 2009
lec 13M.37
Babies & Full Moons
influence of IO’s magnetic
field changes with phases!
--might affect radios in
ambulances, for example
Albert R Meyer, May 4, 2009
lec 13M.38
Babies & Full Moons
So independence of B and F
is actually unclear.
Deciding whether to treat
them as independent is a
matter experiment, not
Mathematics.
Albert R Meyer, May 4, 2009
lec 13M.39
Mutual Independence
events A1, A2,
,An are
mutually independent
iff Pr{Ai  Ai 
1
 Ai } 
2
Pr{Ai }  Pr{Ai }
for all Ai
1
j
2
k
Pr{Ai }
k
2n-(n+1) equations
to check!
Albert R Meyer, May 4, 2009
lec 13M.42
Mutual Independence
choose values v1,v2,
with = probability.
independently
for events [vm = vn]:
any 2 are independent
but [v1 = v2], [v2 = v3], [v1=v3]
not mutually indep.
Albert R Meyer, May 4, 2009
lec 13M.43
Mutual Independence
Events E1, E2, ... are
k-wise independent
iff every subset of k of them
is mutually independent
Albert R Meyer, May 4, 2009
lec 13M.44
Team Problems
Problems
2 4
Albert R Meyer, May 4, 2009
lec 13M.45