COS 341, September 29, 1997Handout Number 5
Math Casino
Let 
. Roulettes in the Math Casino have the following rule.
For each run, a random integer n, 
is uniformly
generated. If 
divides n, you win
$5; otherwise, the $1 bet you place is forfeited. Should you
play the game? One way to make a rational decision is to
calculate the probability p of winning a game, and play the
game if and only if 
. How can we compute p?
By definition p=m/N, where m is the number of integers 
that satisfy divides n''.
Let 
be the set of integers n satisfying 
(i.e., 
)
and j dividing n''. Clearly,
where the term 1 comes from the 
contribution.
Let 
. Starting at 
, every j-th integer in the
range satisfies j dividing n.
Hence,
This shows
It remains to evaluate 
. Using
the equality 
,
we have
From (1) we then have
Hence p=m/N=0.1456 <1/5, and you shouldn't play this game.
This solves the Math Casino Problem.