When probabilities are quoted without specification of the sample space, it could result in ambiguity when the sample space is not self-evident. To avoid this, the sample space can be explicitly made known. The probability of an event A given sample space S, denoted by P(A|S), is nothing but the conditional probability of A given S.
Consider the probability of having the habit of smoking, say, in the whole population of a country. This is the unconditional probability "to be a smoker". The probability that a person is a smoker, given that the sex is "female" (or age is "below 14", or racial group is "white", etc.) are examples of conditional probability.