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 north-east % or south-east 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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