Делать что-либо на авось, надеяться на авось (to do something on the off chance, faith in luck) it means to rely on chance, on luck.
It is quite common to hear among foreigners talks about one of the peculiarities of Russians, I mean how Russian can start a business or anything else without an accurate detailed plan, relaying mostly of fate.
This is partly correct. Some psychologists believe that this feature of many Russians has deep historical roots and is related to the natural conditions that have surrounded our ancestors, their weakness in front of these conditions. In the most parts of our territory, the climate distinguishes itself by an extreme instability: long periods of rain and sudden drop of temperatures during the summer can be replaced by heat and drought, so that the crop was more dependent on the weather that on the labour of the farmer. There was a saying among the people about this: “на авось мужик и хлеб сеет” (man sow the crop and depends on luck).
Russia is covered with vast areas of forests and bogs, therefore, when taking the road, people couldn't be sure that their travels would end up safe, that their horse could go through these obstacles. And thus appeared the expression “авось пойдёт, авось пройдёт, авось получится” (to start a travel relying on chance, to hope for the best along the way, to succeed by luck<lt;/em>).
In modern language, those sentences are not only used when talking about a trip, but with any other things. Numerous examples can be met in Russian literature. N. Gogol wrote in the “Revisor”: “Не знаешь, с какой стороны и приняться. Ну да уж попробовать. Что будет, то будет, попробовать на авось.” (One doesn't know where to begin. But I will try. Come what may, I'll try - hit or miss.)
The older generation can still remember the name of the bag, that could be easily slipped in a pocket, “авоська” (string bag). They appeared not long after the 1917's Revolution because of the terrible deficiency of food. Getting out of the house at the slightest pretext, people took this string bag with them, hoping that on their way, they might find some shops who would sell something.
The origin of the expression “на авось” (by chance) is not clear. Some specialists assume that it is the result of the phonetic contraction of the expression “а вот сейчас = а во се” (and here now, which gives а во се if you take only the firsts letters of each word) which meant “а вот сейчас вдруг случится то, на что надеялись.” (and here now, suddenly happens what you have been hoping for.)
With the development of the society, the dependence of people on external conditions became less important, and consequently new expressions appeared in our language, in which you can feel the condemnation of the faith in the chance. “Кто авосьничает, тот и постничает” meaning that sometimes one starves while believing in faith.
Nowadays, you can read these kind of sentences in newspapers: “В науке ничего не может быть на авось. Всё должно быть строго просчитано и проверено” (in science, nothing can be at random. Everything must be carefully calculated and verified.)
To do something on the off chance, (faith in luck) it means to rely on chance, on luck.