A two-stage batch acceptance rule

In each batch of manufactured articles the proportion of defective articles is $p$. From each batch a random sample of nine is taken and each of the nine articles is examined. If two or more of the nine articles are found to be defective that batch is rejected; otherwise it is accepted. Prove that the probability that a batch is accepted is $(1-p)^8(1+8p)$.

It is decided to modify the sampling scheme so that, when one defective is found in the sample, a second sample of nine is taken and the batch rejected if this contains any defectives. With this exception, the original scheme is continued. Find an expression in terms of $p$ for the probability that a batch is accepted.

For this modified scheme evaluate the average number sampled per manufactured batch over a large number of batches when $p$ has the value 0.1.