What Difference Does Twin-Linked Make?

This posting is inspired by Ron, who asked me the other day: will twin-linked weapons matter?, in relation to my exploration of the central limit theorem posting.

What I'm going to do is to compute the likelihood of hitting with a given ballistic skill (BS), and compare it with the likelihood of hitting if the shooter has a twin-linked weapons. One of the most common examples (especially for myself as a chaos marine player) is that of a marine (BS=4) shooting a bolter, versus a terminator (also BS=4) shooting a twin-linked bolter.

Without Twin-Linked.
This is straight-forward to work out. For BS=4, the marine is hitting on 3+ with a d6. This means that two-thirds of the time a hit is scored (or 66.7% probability).

With Twin-Linked.
Twin-linked weapons allow the re-rolling of the die to hit, if a miss is scored. Hence, the probability of hitting on the second time around is also 66.7%.

So what is the combined probability to hit with BS=4 and a twin-linked weapon? The answer is 66.7%, plus 66.7% of 33.3%. (The 33.3% is the probability that you missed on the first d6 roll). Two-thirds (66.7%) of one third (33.3%) equals 22.2%. Therefore the combined probability is 88.8%. That's almost 9 times out of 10 that a BS=4 marine will hit with a twin-linked bolter. Pretty good.

Below, I've tabulated the twin-linked probabilities for other BS's:

    NOT Twin-  Twin-
BS  Linked     Linked
1    16.7%     30.6%
2    33.3%     55.6%
3    50.0%     75.0%
4    66.7%     88.8%
5    83.3%     97.2%
6    86.1%     97.2%
7    88.9%     97.2%
8    91.7%     97.2%
9    94.4%     97.2%
10   97.2%     97.2%

Note that on page 18 of the rulebook, the twin-linked rule gets preference over the chart for BS=6 or greater (dice can only ever be re-rolled once!). This explains why all probabilities to hit with a twin-linked weapon at above BS=4 are all exactly the same. Therefore, if the shooter has BS=10, then they don't exactly need a twin-linked weapon.

Finally, I should also add that twin-linked flamers (and the like) don't roll to hit anyway. If twin-linked, they re-roll to wound (rather than to hit) and can re-roll armour penetration against vehicles. Twin-linked blasts are a different story (re-rolling the scatter die), and that's a story for another day.