1
From Urns to Coupons
•
“Coupon Collecting” is classic probability problem
There exist
N
different types of coupons
Each is collected with some probability
p
i
(1 ≤
i
≤
N
)
•
Ask questions like:
After you collect
m
coupons, what is probability you
have
k
different kinds?
What is probability that you have ≥ 1 of each
N
coupon
types after you collect
m
coupons?
•
You’ve seen concept (in a more practical way)
N
coupon types =
N
buckets in hash table
collecting a coupon = hashing a string to a bucket
Digging Deeper on Independence
•
Recall, two events E and F are called
independent if
P(EF) = P(E) P(F)
•
If E and F are independent, does that tell us
anything about:
P(EF  G) = P(E  G) P(F  G),
where G is an arbitrary event?
•
In general, No!
Notso Independent Dice
•
Roll two 6sided dice, yielding values D
1
and D
2
Let E be event: D
1
= 1
Let F be event: D
2
= 6
Let G be event: D
1
+ D
2
= 7
•
E and F are independent
P(E) = 1/6,
P(F) = 1/6,
P(EF) = 1/36
•
Now condition both E and F on G:
P(EG) = 1/6,
P(FG) = 1/6,
P(EFG) = 1/6
P(EFG)
P(EG) P(FG)
EG and FG
dependent
•
Independent events can become dependent by
conditioning on additional information
Do CS Majors Get Less A’s?
•
Say you are in a dorm with 100 students
10 of the students are CS majors:
P(CS) = 0.1
30 of the students get straight A’s: P(A) = 0.3
3 students are CS majors who get straight A’s
o
P(CS, A) = 0.03
o
P(CS, A) = P(CS)P(A), so CS and A are independent
At faculty night, only CS majors and A students show up
o
So, 37 (= 10 + 30 – 3) students arrive
o
Of 37 students, 10 are CS
P(CS  CS or A) = 10/37 = 0.27
o
Appears that being CS major lowers probability of straight A’s
o
But, weren’t they supposed to be independent?
In fact, CS and A conditionally dependent
at faculty night
Explaining Away
•
Say you have a lawn
It gets watered by rain or sprinklers
P(rain) and P(sprinklers were on) are independent
Now, you come outside and see the grass is wet
o
You know that the sprinklers were on
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 Spring '09
 Probability theory, CDF, CS majors

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