Assuming equal base-probability, the probability of the exact 4-base sub-sequence 'ATTG' is 0.25 ^ 4. In a 20 base sequence you can fit a 4 base sequence in 17 places. So the expected number of occurrences is 0.25 ^ 4 * 17 == 0.06640625

You can make this generic:

def prob(seq_len, subseq_len):
    if subseq_len > seq_len: return 0
    places = seq_len - subseq_len + 1
    prob = 0.25 ** subseq_len
    return prob * places

print prob(20, 4)

translating to R is up to you.