At a carnival you win a prize if you get a head, you must first choose a coin. There is a fair and a biased coin, while choosing each coin is equally likely, the biased coin has a $78 \%$ of landing tails. What is the probability of choosing the biased coin if you won a prize?

Step 1 ：We are given a problem of conditional probability. We are asked to find the probability of choosing the biased coin given that we won a prize. We know that we win a prize if we get a head.

Step 2 ：We can use Bayes' theorem to solve this problem. Bayes' theorem states that \(P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)}\). In this case, event A is choosing the biased coin and event B is winning a prize.

Step 3 ：We know that \(P(A) = 0.5\) (probability of choosing the biased coin), \(P(B|A) = 0.22\) (probability of winning a prize given that we chose the biased coin), and \(P(B)\) is the total probability of winning a prize.

Step 4 ：To find \(P(B)\), we need to consider both cases of choosing the fair coin and the biased coin. If we choose the fair coin, the probability of winning a prize is 0.5. If we choose the biased coin, the probability of winning a prize is 0.22. Since choosing each coin is equally likely, \(P(B) = 0.5 \cdot 0.5 + 0.5 \cdot 0.22\).

Step 5 ：Once we have all these values, we can substitute them into Bayes' theorem to find the answer.

Step 6 ：\(P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)} = \frac{0.22 \cdot 0.5}{0.36} = 0.3055555555555556\)

Step 7 ：Final Answer: The probability of choosing the biased coin if you won a prize is approximately \(\boxed{0.306}\).