Santa's workshop makes 10 different types of sack for Santa and his elves, types \(A_{1}, A_{2}, \ldots, A_{10}\). The number of \(A_{x}\) type sacks made are \(17 \times x\). In other words, there exist \(17 A_{1}\) sacks, \(34 A_{2}\) sacks, etc. The proportion of soft gifts depends on the type of sack they is in. Type \(A_{1}\) contains \(1 \%\) soft gifts, \(A_{2}\) contains \(4 \%\), and in general, sacks of type \(A_{x}\) contain \(x^{2} \%\) soft gifts. Each sack contains one million gifts. Your elf gets one such sack, picked at random.

- What is the probability that your elf got a sack of type \(A_{x}\) ?

- The elf pulls out two random gifts for you. Both are soft. Use this information to update the probabilities of which type of sack your elf picked.

- What is the probability that your elf has a bag of type \(A_{8}\) ?