Here's the full calculation.
Probability of Sheffield Derby = P(sd)
Probability of Sheffield Derby as Tie 1 P(t1)... etc
Probability of drawing Sheffield club = P(s)
Probability of drawing any other club = P(o)
We'll take the probability of each ball in order giving us the outcome of a Sheffield Derby until we get the two clubs out together, so P(t) = P(b1) x P(b2) x P(b3), etc. We could do all 16 balls each time, but once we have both Sheffield clubs out then we're just left with P(o) = 1 for the remaining balls.
Here's the really tedious bit:
P(t1) = P(s) x P(s) = 2/16 x 1/15 = 2/240 = 1/120
P(t2) = P(o) x P(o) x P(s) x P(s) = 14/16 x 13/15 x 2/14 x 1/13 = 364/43,680 = 1/120
P(t3) = P(o) x P(o) x P(o) x P(o) x P(s) x P(s) = 14/16 x 13/15 x 12/14 x 11/13 x 2/12 x 1/11 = 48,048/5,765,760 = 1/120
P(t4) = P(o) x P(o) x P(o) x P(o) x P(o) x P(o) x P(s) x P(s) = 14/16 x 13/15 x 12/14 x 11/13 x 10/12 x 9/11 x 2/10 x 1/9 = 4,324,320/518,918,400 = 1/120
(the multiplied fractions are getting silly now, so I'll skip to the simplest fraction, and I think we can see a pattern emerging)
P(t5) = P(o) x P(o) x P(o) x P(o) x P(o) x P(o) x P(o) x P(o) x P(s) x P(s) = 14/16 x 13/15 x 12/14 x 11/13 x 10/12 x 9/11 x 8/10 x 7/9 x 2/8 x 1/7 = 1/120
P(t6) = P(o) x P(o) x P(o) x P(o) x P(o) x P(o) x P(o) x P(o) x P(o) x P(o) x P(s) x P(s) = 14/16 x 13/15 x 12/14 x 11/13 x 10/12 x 9/11 x 8/10 x 7/9 x 6/8 x 5/7 x 2/6 x 1/5 = 1/120
P(t7) = P(o) x P(o) x P(o) x P(o) x P(o) x P(o) x P(o) x P(o) x P(o) x P(o) x P(o) x P(o) x P(s) x P(s) = 14/16 x 13/15 x 12/14 x 11/13 x 10/12 x 9/11 x 8/10 x 7/9 x 6/8 x 5/7 x 4/6 x 3/5 x 2/4 x 1/3 = 1/120
P(t8) = P(o) x P(o) x P(o) x P(o) x P(o) x P(o) x P(o) x P(o) x P(o) x P(o) x P(o) x P(o) x P(o) x P(o) x P(s) x P(s) = 14/16 x 13/15 x 12/14 x 11/13 x 10/12 x 9/11 x 8/10 x 7/9 x 6/8 x 5/7 x 4/6 x 3/5 x 2/4 x 1/3 x 2/2 x 1/1 = 1/120
P(sd) = P(t1) + P(t2) + P(t3) + P(t4) + P(t5) + P(t6) + P(t7) + P(t8)
P(sd) = 1/120 + 1/120 + 1/120 + 1/120 + 1/120 + 1/120 + 1/120 + 1/120
P(sd) = 8/120
P(sd) = 1/15
If anyone wants to argue further that it is anything other than 1/15 then there really is no hope.