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This question relates to the 2nd edition of Probability and Computation by Mitzenmacher and Upfal.

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The authors then use the Law of Total Probability enter image description here

My question concerns the line that begins with $\leq$. Shouldn't that be replaced with $=$ because the authors proved that the event $\mathbf{AB} \bar{r} = \mathbf{C}\bar{r}$ is equivalent to the event $$r_{1} = - \frac{\sum_{j=2}^n d_{1j} r_{j}}{d_{11}}$$ ?

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  • $\begingroup$ Don't use images as main content of your post. This makes your question impossible to search and inaccessible to the visually impaired; we don't like that. Please transcribe text and mathematics. You can use LaTeX. $\endgroup$
    – D.W.
    Oct 31 '21 at 23:04
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I'm assuming that all matrices in question are $n \times n$.

Suppose that $A=B=0$ are zero matrices and $C=I$ is the identity matrix. Then $$ \Pr[ABr = Cr] = \Pr[r = 0] = 2^{-n}. $$

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