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Make a markov system with two states:

1:not crashed; 2: crashed (an absorbing state)

With this numbering, we get the following transition matrix

9,999/10,000	1/10,000
0	1

We must calculate the number of steps to absorption. To do this, we need the fundamental matrix. First, S is the 11 matrix

S = [9,999/10,000] .

Thus,

Q	=	(I-S)^-1
	=	[1/10,000]^-1
	=	[10,000]

Since the 1, 1 entry is 10,000, it follows that one can expect to be in state 1 10,000 times on average before absorption (including the first trip). In other words, you can expect a crash on the 10,000th trip (on average).

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