Passing Down Hidden Abilities?

Question

So suppose I have a female Snivy with Overgrow and I breed it with a male Eevee that has the hidden ability anticipation. Is there any chance at all that I will get a contrary Snivy.

Answer 1

Quite simply, no. It's all about the female. Only when the female has its hidden ability is there a chance (not a guarantee) to pass it down. The only exception is when breeding with a ditto. In this scenario, only the male may pass down a hidden ability, an imposter ditto makes no difference.
EDIT

Bulbapedia

Starting in Black and White, female Pokémon with Hidden Abilities have a 60% chance of passing on the Hidden Ability to their offspring when bred with a male Pokémon from the same Egg Group. In Generation VI, this was extended so that male and genderless Pokémon with a Hidden Ability can also pass it down if they are bred with Ditto. The chance of the offspring having either non-Hidden Ability is 20% each.

Serebii

The most notable change is that Hidden Abilities, which previous required you to have the mother with the Hidden Ability, can now be bred without the mother having it. If the father has got the Hidden Ability, there is a small chance that the offspring will also have their Hidden Ability. However, this only works if you are breeding with a Ditto. This is even more useful considering Hidden Abilities are more easily found via Hordes and Friend Safari.

sources
http://bulbapedia.bulbagarden.net/wiki/Ability#Hidden_Abilities
http://www.serebii.net/xy/breeding.shtml

Comments

This is no longer strictly true.  In X/Y/OR/AS (and presumably Sun/Moon going forward) the male gives a 20% chance of passing it's Hidden Ability when paired with a female of the same specie.  The female has an 80% chance if she has her Hidden Ability.

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If you're going to tell some one they're wrong, you had better have evidence to back it up

Answer 2

In order to hatch a Pokemon with it's Hidden Ability, one or both of the parents must already have that Hidden Ability. When pairing Pokemon of different species, the female will decide which specie the baby is, which then means in that case the female must have the Hidden Ability for the baby to have it.