MATH 145: HOMEWORK 2
ANDREW BERGET
As before, do not hand in [bracketed] problems. This homework is due on Wednesday January 20.
Problem A.
We toss a fair coin
n
times and get get
h
heads and
t
tails where
h > t
are ﬁxed integers.
Your goal in this problem is to prove that
The probability that, as we toss the coin, the number of heads is
always larger than the number of tails is equal to
(
h

t
)
/
(
h
+
t
)
.
For example, if
h
= 3 and
t
= 2 then
the sequences of tosses
HHTHT
satisﬁes our condition, but
HTHHT
does not since at the second toss the
number of tails equals the number of heads.
(1) A walk on the grid of points with integer coordinates that only uses steps that are northeast
%
or
southeast
is called a
diagonal lattice path
. How many diagonal lattice paths are there from (0
,
0)
to (
u,v
)? (This is essentially solving Problem 31 in your book.)
You can check that your answer is correct by showing that the number of lattice paths from (0
,
0)
to (2
,
2) is 1 and the number of such paths from (0
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 Winter '07
 Peche
 Combinatorics, Integers, lattice paths

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