Lower Bounds on Full Rank Probability in Random Matroids

Let $M$ be a matroid on the set $E$ whose elements are missing jointly independently with probability $Q$. The probability $P(M,q)$ of full rank of a random matroid $(M,q)$ is the probability that the random set $(E,q)$ contains a base of this matroid. It is proved that every partition of $E$ into independent sets of $M$ that contains a base of $M$ generates an effectively computable lower estimate of $P(M,q)$ in terms of the conjugate partition. A partition that yields the best such estimate is determined.