Problem 1. A couple decides to have children until they have both a boy and a girl. What is the expected number of children that they'll end up with? Assume that each child is equally likely to be a boy or a girl and genders are mutually independent Solution. There are many ways to solve
1 Graphs and Trees The following two definitions of a tree are equivalent Definition 1: A tree is an acyclic graph of n vertices that has n-1 edges Definition 2: A tree is a connected graph such that Vu, v E V, there is a unique path connecting u to u. In general, when we want to show the equivalence of two definitions, we must show
1 Unstacking Here is another wildly fun 6.042 game that's surely about to sweep the nation! You begin with a stack of n boxes. Then you make a sequence of moves. In each move, you divide one stack of boxes into two nonempty stacks. The game ends when you have n stacks, each containing a single box You earn points for each move; in particular, if you divide one stack of height a b into two stacks with heights a and b, then you score
Random walks 1 Random walks a drunkard stumbles out of a bar staggers one step to the right, with a canal lies y steps to his right. Thi I equal p second, he either staggers one step to the left or probability. His home lies r steps to his left, and everal natural questions, including 1. What is the probability that the drunkard arrives safely at home instead of falling into the canal? 2. What is the expected duration of his journey however it ends? The drunkard's meandering path is called a random walk. Random walks are an im- portant subject, because they can model such a wide array of phenomena. For example
Check n. A written order to a bank to pay the amount specified from funds on deposit;, a draft. 支票给银行从存款中支付具体款项的书面支票;汇票 Issue v.tr. To circulate or distribute in an official capacity:
