science of board games

There is an option in Twilight Imperium 3rd Edition: you can take fighters, which are throwaway units, and have them do a "trench run" on a "war sun" (à la Star Wars). At first glance, this might be a good idea, as fighters are dirt cheap and not very powerful, while war suns are the most expensive and destructive ships. So, how good or bad of an idea is it really?

The rules state that you roll a 10-sided die for each of the fighters making the run: on a 9 or 10, they live through the first barrage. The fighters that live must then roll the die again, getting a perfect 10. This math is easy to do: you have a 2%, or 1/50, chance of killing it. Therefore you must have 50 fighters to get guaranteed destruction, right?

Nope. And with dice, there's no such thing as "guaranteed" anything. The best we can hope for is to be more than likely able to defeat that War Sun.

To start this calculation, we think about how likely it is that the war sun survives a single attack? We know from above that this is 49/50, or 0.98. Well, how likely is it to survive after two attacks? Take 0.98 × 0.98 = 0.9604. After three attacks? 0.98³ = 0.941192.

We want to know when we will have a better than half chance, i.e., 0.50. So we solve the following equation:

0.98^x = 0.50

Using some elementary algebra and logarithms, the answer will be log(0.50) / log(0.98) = 34.309, or about 35 fighters.

While 35 is better than 50, that's still an awful lot of fighters. Fighters, by default, hit on 9's and 10's, so a fleet of 35 of them would do about 7 damage, which is no number to sneeze at: it would take down many moderate sized enemy fleets on the first barrage, and it would be unlikely that the enemy could do 35 damage every turn, so the fighters would just keep nailing the other player.

Unless you have just a handful of fighters and not a lot else anywhere, a trench run ain't such a good idea. And if someone wants to attack your War Sun with less than 35 fighters, let them. Even with a more normal and strong-sized fleet of 10 fighters, a War Sun has an 81.7% chance of survival, and will be guaranteed to kill 80% of the attacking fighters.