I had a character with an axe that had a 67% chance of hitting and would strike twice (each hit has the 67% hit probability). I had three turns to hit a character before they escaped with a valuable item. It would take two hits to defeat this enemy.

Now, Fire Emblem 3 Houses lies with its % reporting. Percentages above 50% are actually higher than reported and percentages lower than 50% are lower than reported. Without going into too much detail, a reported 67% hit chance is actually 78.55%.

So, to not defeat the enemy in time, I needed every attack to miss or five of the attacks to miss. The probability for every attack missing is

1) (1 - 0.7855)⁶

If we have a 78.55% chance to hit then we have a 100% - 78.55% to miss. We multiply this probability for each miss.

Meanwhile, the probability for five misses and one hits is
2) 6 * (0.7855)¹ * (1 - 0.7855)⁵

We have the probability of missing five times and the probability of hitting once. Now, why the multiplicative factor of six? That's because the single attack that hits could be the first, second, third, fourth, fifth, or sixth swing, and we must account for every case.

The total probability of not defeating the enemy is the sum of the two cases.

(1 - 0.7855)⁶ + 6 * (0.7855)¹ * (1 - 0.7855)⁵ = 0.00223750036

which is about a 0.22% chance. The probability of defeating the enemy was 0.99776249963 or about a 99.776% chance. Fuck me