How to get the Astronomically Low Odds achievement?

The best way to do this achievement is to find an encounter where the enemy has no way of hurting you. This is the same technique used to train crews but is much more difficult in this case as it can only be performed late, after you have full engines.

There is, however, one ship that is perfect for this- the Rebel Flagship.

Enter the fight with fully upgraded engines, at least 6 shields (3 layers), level 3 piloting system, a teleporter and a decent boarding crew. Begin the fight by boarding and destroying the missile launcher, the ion blaster and the halberd beam. Leave the lasers (middle-left) alone.

You now have an enemy armed only with a burst laser 2. It is impossible for them to break your 3 shields, repair their guns, flee or do anything at all. All you need to do is man the engines and piloting and wait for the 5 hits in a row- you don't even need to watch it.

The chance to get this is not really astronomically low. Your dodge chance to complete this achievement would be 45%, 35% from the engines and another 10% from your pilot as long he gained no levels. This leaves 55% chance the attack hits. 0.55*0.55*0.55*0.55*0.55 = 0.05 = 5%.

You need to have a active pilot. Since pilots level up and increase the dodge rating you should try to get the autopilot system or have only a active pilot when the opponent is using missiles.

Obviously you should not build defense drones since a lower amount of missiles have the chance to be dodged as the drones shooting them happens first. Also it does not say you have to man

By upgrading the engines as fast as possible you increase the amount of possibilities this could happen so effectively raises the chance over time.

You should try to find ships that fire many bursts of low damage. Also a hull repair drone helps you last longer while trying to get this achievement.

Further more there is a possible bug with this achievement.

Taken from FTL wikia:

As of Steam Version 1.03.3 for Windows, I was unable to get this achievement without also upgrading the Piloting Sub-System to level 3. This doesn't seem like it should be necessary, but I let the game run for roughly 5 hours against an opponent with a Level 3 Burst Laser (5 shots), watched combat for about 30 minutes and witnessed 5 non-dodged shots in a row many times, assumed many others happened over the course of 4 to 5 hours while I was not watching, and could not get this achievement. When I upgraded the Piloting Sub-System to level 3, I got the achievement a few minutes later. This could very well be a bug, but if you are having difficulty, you may want to try upgrading the Piloting Sub-System.

You only need to fully upgrade the engines and man the piloting station with an inexperienced crew member. That's a 35%+5% = 40% dodge change.

Engines don't need to be manned. Piloting station does not need to be upgraded. Only using the autopilot does not work (assumption).

Since you have a 60% change of getting hit, the chance to get hit 5 times in a row is 7.8%.

I got it almost by mistake in the first fight after upgrading the engines. There was an asteroid field and the enemy ship had two combat drones.