 # Gears of War: Comparing attacks

In my last post I looked at the chance of killing an enemy with the chainsaw, which came to 52%.

In this post I will do some more calculations to see how other attacks compare to this.

When deciding whether or not to use the chainsaw, you are most likely going to want to compare it to the normal and overkill Lancer Rifle attacks, which are 2 and 4 dice respectively. For example, if you want to save your last ammo token, should you use the chainsaw or a normal attack?

The attack die has the following sides: 1 omen, 1 two-hit, 3 one-hit and 1 blank. The lancer rifle treats an omen as a one-hit.

The attack die therefore has six outcomes (2 + 1 + 1 + 1 + 1 + 0) which total 6 hits, so the average is 1 hit. For two dice, the average is 2 hits.

If you break it out, there are 36 possible results from rolling two dice.

They are: 1 in 36 results is four hits, 8 in 36 are three hits, 18 in 36 are two hits, 8 in 36 are one hit, and 1 in 36 is no hits. If you add all these up (4 + 24 + 36 + 8 = 72) and divide by 36, you get an average of two hits.

Now lets look at the defence die. This has the following sides: 3 blanks, 2 one-shield, and 1 two-shields. This totals 4 shields over 6 different outcomes, which is an average of two-thirds, or 0.66.

Wretches and drones both have 1 defence. If we are comparing to the chainsaw, then we must be in the same area and so cover does not count, so we roll only one defence die.

A very simple analysis says that, on average, a point-blank normal lancer shot against a wretch or drone will do 2 hits minus 0.66 shields = 1.33 damage.

Wretches have 2 health and drones have 3 health, so this is not enough to kill either one. This means the chainsaw is better than a point-blank normal lancer attack.

What about a point-blank overkill attack? This uses four dice, which has an average result of 4 hits. So the average damage will be 4 minus 0.66, which is 3.33 and is enough to kill both.

That extra .33 suggests that it is a more than 50% likely to do 3 damage, but it would be nice to know if it is definitely more than the 52% of the chainsaw.

The arithmetic does get a bit mad. If you aren’t interested, just skip to the end. Also, I can’t guarantee these results are 100% correct, they are just my quick back-of-a-napkin notes. The wretch and drone locust cards show their defence (in red square) and health (the red number next to it).

To do at least 3 damage, you need to roll at least 4 hits.

With four dice, there are 1296 outcomes (6x6x6x6).

A total of 0 hits can only happen 1 way (all four dice are blanks).

A total of 1 hit can happen 16 ways. This is three zeroes and a one. The one can be on any of the 4 dice, and there are 4 ones on each die, so that is 4×4=16.

Exactly 2 hits can happen 100 ways. This is either (a) two zeroes and two ones, or (b) three zeroes and a two.

For (a), there are 16 ways to have two ones on two dice (4 ones on each of two dice is 4×4=16 combinations). Then there are 6 combinations of the four dice with two zeroes and two ones (1100, 1010, 1001, 0110, 0101, 0011). So that is 16 x 6 = 96.

For (b), there are only 4 ways to have a single two (on one of 4 dice).

This gives a total of 96 + 4 = 100 ways to get 2 hits.

For exactly 3 hits we must have either (a) one zero and three ones, or (b) two zeroes and a one and a two.

For (a), there are 64 ways to have three ones on three dice (four ones on each dice = 4x4x4), and 4 combinations of a zero and three ones on four dice (0111, 1011, 1101, 1110), giving 64×4 = 256.

In case (b), there are 12 combinations of one and two out of four dice (1200, 1020, 1002, 0120, etc), and 4 ones on the dice showing a one, which is 12×4=48.

This gives a total 256 + 48 = 304 ways to get exactly 3 hits.

Now we have enough information to figure out some probabilities.

The chance of scoring at least two hits is the opposite of only scoring 0 or 1. There are 17 ways to score 0 or 1 hits (1+16). This leaves 1279 ways to score 2 or more hits (1296-17). The probability is 1279/1296 = 0.99 = 99%.

Similarly, the chance of scoring at least 3 hits is the opposite of only scoring 0, 1 or 2, which is (1296-1-16-100)/1296 = 0.91 = 91%.

Finally, the chance of scoring at least 4 hits is the opposite of only scoring 0, 1, 2 or 3, which is (1296-1-16-100-304) /1296 = 0.68 = 68%.

So, the point-blank overkill lancer attack kills a drone 68% of the time, and kills a wretch 91% of the time. Definitely much better than the chainsaw. If you absolutely, positively have to kill every last locust in the room, accept no substitutes.