Coupon collector’s problem
CS658
Po-Ching Liu
Question
• A person trying to collect each of b different
coupons must acquire approximately x
randomly obtained coupons in order to
succeed.
• find x
Balls and bins
• 1st question:
If n identical balls are tossed randomly into b
bins, then how many balls will fall in a given bin?
n
• Ans:
b
 the probabilit
y that a tossed ball lands in any given bin is
1
b
the expected
number
of n balls
 n
1
b

n
b
.
,
Second
• Second question:
Find the expected number of balls we need to
toss (one ball for each toss) until a given bin
contains a ball.
The number of bins = b
• Ans: E = b
proof
• p=pro(success) = 1/b=1-q
• q=pro( without success)=1-1/b=1- p
• E[# of tosses until a given bin contains a ball]
E  1  p  2  qp  3  qqp  4  qqqp  
 p [1
 2 q  3 qq  4 qqq   ]
 p [( 1

q
 q (1
 qq 

qqq   )
q 
qq   )
 qq (1 
q  )
 qqq(1   )
 ]
 (1

 E  p(
 p(
1
1 q
 qq
q
1
1 q
)(
 qqq   ) 
1
1 q

)( 1  q  qq  qqq   )
1
1 q
)  p
1
p

1
p

1
p

1
b
1
( )
b
1
p
Third
• the third question:
Find the expected number of balls we need to
toss (one ball for each toss) until every bin
contains at least one ball?
The number of bins = b
• Ans: E = b(lnb+O(1))
proof
• There are b stages
• The ith stage consists of the tosses after
the (i-1)th hit until the ith hit.
• how many balls do we have to toss in order to
move from the (i-1) th stage to the ith stage
 there are (i-1) bins that contain balls and
b-(i-1) empty bins.
for each toss in the ith stage, the
probability of obtaining a hit is (b-i+1)/b.
proof
b
• E[# of tosses in the ith stage]= b  i  1
st to the bth stage]
• E[total
#
of
tosses
in
the
1
b
b
1
1
1
1 1
= 
 b( 

   )
b  i 1
b b 1
1
1
1
1 1
( 

   )
b b 1 b  2
2 1
1 1 1
1
1
 (    
 )
1 2 3
b 1 b
i 1
b

b2
2
1
1
i
i 1
b


x 1
1
x
b
dx  1 
1
i
i 1
b


x 1
1
x
dx  [ln x ]1  c  ln b  c
b
c  constant
proof
b
b
 b 1 i
i 1
 b(
1
b

1
b 1

1
b2
 
1
1
 )
2 1
 b (ln b  c )  b (ln b  O (1))
• We find the two following questions are the same:
• Q:Find the expected number x of balls we need to toss (one
ball for each toss) until every bin contains at least one ball?
The number of bins = b
• Q: A person trying to collect each of b different coupons
must acquire approximately x randomly obtained coupons
in order to succeed.
 x  b( ln b  O( 1 ))
Code by java compile: javac CouponCollector.java
Run: java CouponCollector
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public class CouponCollector {
public static void main(String[] args) {
int N = 50; // number of different card types
boolean[] found = new boolean[N];
// found[i] = true ==> if card i has been collected ,
int cardcnt = 0;
// total number of cards collected
int valcnt = 0;
// number of distinct cards
false ==> the new type of card
// repeatedly choose a random card and check whether it's a new one
while (valcnt < N) {
int val = (int) (Math.random() * N);
// random card between 0 and N-1
cardcnt++;
// we collected one more card ==>total number +1
if (!found[val]) valcnt++;
// it's a new card type
found[val] = true;
// update found[]
}
// print the total number of cards collected
System.out.println(cardcnt);
}
}
references
• Thomas H. Cormen,Charles E. Leiserson,
Ronald L. Rivest, Clifford Stein(1990),
Introduction to Algorithms, The MIT Press,
Cambridge, Massachusetts London, England,
pp. 109-110.
• http://wwwstat.stanford.edu/~susan/surprise/Collector.ht
ml