The sequence GAATTC is a palindromic sequence, meaning it reads the same forward and backward. Therefore, the odds of EcoRI cutting a random stretch of 6 nucleotides is the same as the probability of randomly selecting the sequence GAATTC within a 6-nucleotide stretch.
Each nucleotide in DNA can be one of four possibilities: A (adenine), T (thymine), G (guanine), or C (cytosine). Therefore, the probability of randomly selecting GAATTC within a 6-nucleotide stretch is (1/4) * (1/4) * (1/4) * (1/4) * (1/4) * (1/4) = (1/4)^6.
Simplifying this expression, we have (1/4)^6 = 1/4096.
Therefore, the odds that EcoRI will cut a random stretch of 6 nucleotides is approximately 1 in 4,096.
Therefore, the closest option is "approximately 1 in 4,000".