Posterior for a Double-Headed Coin

A box contains 100 coins: 98 fair coins, 1 double-headed coin (heads on both sides), and 1 double-tailed coin (tails on both sides). You pick a coin uniformly at random and flip it. 1. The first flip shows heads. Compute $P(\text{double-headed} \mid \text{heads})$, $P(\text{fair} \mid \text{heads})$, and $P(\text{double-tailed} \mid \text{heads})$. 2. You flip the same coin again and it shows heads again. Update your posterior: compute $P(\text{double-headed} \mid \text{two heads})$.