There's a significant amount of intuiting you can do to reduce each of A, B, C, and D, but given the fact that the soft answers give over 56 thousand possible answers that map into less than 10 thousand, the number theory of the problem naturally lends itself to a lot of collision, complicating the probabilities. Thus, a brute force answer lends itself well to this.
Brute force answer:
7101 with 54 possible combinations of a,b,c, and d mapping to it. Not taking into account any of the "soft" figures like birthdays or letter frequencies.
All in all:
This gives you a 1 in 1040 chance of opening the safe, which is much improved over 1 in 9999.
A has a range of 1-9 assuming people don't start numbers with 0. B has an obvious range of 1-26. C has a range of 1-20 with a maximum probability of being 1. D has an obvious range of 1-12. From there, you can brute force all 56160 possibilities in less than a second using python.
If after multiplying the soft probabilities together you get a better ratio than 1/1040, you're better off with the soft answer. Otherwise, your best bet is to use the brute force answer.
5203 which gives you a .1156% chance of opening the safe
Thanks to @kayzeroshort for this one. According to Benford's Law, 1 is the most likely with a probability of about 30%.
E is the most common letter in non-technical English with a probability of 11.16%, which might possibly be shifted by the fact that it's a VCR manual.
The probability of seperation is highest for 1 occurring 38 of the 190 times, giving it a probability of .2
The least birthed month is February with about a .89 probability below the median. Combined with a birthday paradox calculation that nobody was born in February (probability .4189), this gives D as 2 with a chance of 1/.89 * .4189 = .46644
Then you just math the rest.
So your chances in all are:
.3 * .1116 * .2 * .46644 = ~.00321 or a .3121% chance of it working. It's slightly better than brute force assuming all assumptions hold up. We made assumptions that the VCR technical manual was a reasonable sample of modern English (and that it was in English in the first place), and that all of John's friends were American and were born in America fairly recently.
Because A, B, C, and D mix to form the 4 digits, doing a straight multiplication to get probabilities is a little disingenuous. Additionally, 5203 can only be made in 5 combinations of A, B, C, and D, compared to 54 from 7101 from the brute force.