Final answer:
The average distance between restriction enzyme sites of Eco RI in a random DNA sequence is calculated to be once every 4096 base pairs, as it depends on the probability of the specific 6-base pair sequence occurring, which is not listed in the provided options.
Step-by-step explanation:
The average distance between restriction enzyme sites in a random DNA sequence depends on the frequency with which a particular sequence occurs. Since type-II restriction endonucleases, like Eco RI, recognize specific palindromic nucleotide sequences, the average distance between these sites can be calculated based on probability. The sequence recognized by Eco RI is 'GAATTC', which is a 6-base pair sequence. In order to calculate the average distance, we must consider that each of the 4 nucleotides (A, T, G, C) has an equal chance of occurring at each position, making the occurrence of any specific nucleotide to be 1 in 4 (1/4). The probability of finding a specific 6-base pair sequence like 'GAATTC' is (1/4)^6, as each base pair is independent. Therefore, the sequence should occur once every 4^6 or 4096 bases. In terms of base pairs, this translates to an average distance of 4096 base pairs between each Eco RI site. This is not one of the options provided, so the answer would be none of the above if the options a) 50 bp, b) 100 bp, c) 150 bp, and d) 200 bp are the only choices given.