 Solution

For independent equiprobable bases there are $4^3=64$ codons, three of them stops, so $p_{\mathrm{stop}}=3/64$. In a fixed reading frame, and conditional on starting translation, surviving at least $L$ successive non-stop codons has probability $\boxed{(61/64{)}^L}$. For 300 it is about $5.6\times 10^{-7}$. Strictly “exceeding 300” requires 301 non-stop codons, giving $(61/64{)}^{301}$. A specified start codon, codon bias, correlations or looking for a long sequence anywhere in a large genome changes the sampling problem and cannot be silently folded into this single-frame probability.