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Let each element of vector $x_{n \times 1}$ be a random variable which could be equal to one or zero with probability $\frac{1}{2}$. We have $k$ such vectors $x_1,\dots,x_k$. Assuming all operations are over GF(2), what is the probability that the base vector $e_1=[1,0,0,\dots,0]^T$ is spanned by vectors $x_1,\dots,x_k$. This problem is closely related to the one in

Expected number of random binary vectors to make matrix of order n

The difference is we are only interested in the probability of spanning one specific vector. Also, what is the average value of k for which $e_1$ is spanned with probability one?

mhsnk
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  • Here’s what I think is a sketch of an answer to your question: Compute the probabilities $P(k,r)$ that the rank of the span of the $x_i$ is $r$. (This might help you do that: http://math.stackexchange.com/q/324260.) Then note that probability that $e_1$ is in a subspace $V$ given only that the rank of $V$ is $k$ is $\frac{2^k-1}{2^n-1}$ (the same as the probability that any specific nonzero vector is). Then the probability you seek is $\sum_{r=0}^n P(k,r)\frac{2^k-1}{2^n-1}$. – Steve Kass Dec 19 '15 at 20:01
  • Never mind! I was able to open the link. Thank you for your help. It solves the problem. – mhsnk Dec 19 '15 at 20:20
  • Ah, the period at the end of the sentence became part of the link in my comment. For anyone else having trouble, go to http://math.stackexchange.com/q/324260 . – Steve Kass Dec 19 '15 at 21:05

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