"You bet your ass!" = archaic English, which can be losely translated to: "betting one's most precious possession". In ye olde times, a donkey, or an "ass", was worth more than a woman. In fact, one could trade one ass for at least two wifes. Asses were used as multifunctional units, being capable of transporting equipment or people, pushing/pulling equipment forward, kicking the buttocks of that annoying neighbour of yours who keeps stealing your turnips. They were not as dim as horses, and therefore very useful for people having to deal with dangerous situations (for example, moving along a steep cliff): a horse might just walk off at a random point in time, an ass would never. Additionally, for some people, they could even fulfill the roles of women. To conclude: an ass was a much priced possession, and "to bet one's ass" was not to be done in vain.