Step 1: Define the problem and the given information
We have two coins: one fair (50% heads, 50% tails) and one trick coin that always lands on heads. We pick a coin at random, flip it once, and it comes up heads. We need to find the probability that we picked the trick coin.
Step 2: Calculate the probability of picking each coin
Since we pick a coin at random, the probability of picking the fair coin is 1/2, and the probability of picking the trick coin is also 1/2.
Step 3: Calculate the probability of getting heads for each coin
The probability of getting heads with the fair coin is 1/2, and the probability of getting heads with the trick coin is 1.
Step 4: Apply Bayes' theorem to update the probability
We can use Bayes' theorem to update the probability of picking the trick coin given that we got heads. The formula is P(trick coin | heads) = P(heads | trick coin) * P(trick coin) / P(heads).
Step 5: Calculate the probabilities
P(heads | trick coin) = 1, P(trick coin) = 1/2, P(heads) = P(heads | fair coin) * P(fair coin) + P(heads | trick coin) * P(trick coin) = (1/2) * (1/2) + 1 * (1/2) = 3/4.
Step 6: Plug in the values into Bayes' theorem
P(trick coin | heads) = 1 * (1/2) / (3/4) = 2/3.
The final answer is: $\boxed{\frac{2}{3}}$
Note: This result makes sense because getting heads gives us more information about the coin, and since the trick coin always lands on heads, it's more likely that we picked the trick coin given that we got heads.