Step 1 :Given values are: the percentage of companies listed on Exchange 1, \(p_{\text{exchange1}} = 0.4446\), the probability of a stock advancing on Exchange 1, \(p_{\text{advance | exchange1}} = 0.2648\), the percentage of companies listed on Exchange 2, \(p_{\text{exchange2}} = 0.5554\), and the probability of a stock advancing on Exchange 2, \(p_{\text{advance | exchange2}} = 0.2077\).
Step 2 :First, calculate the total probability of a stock advancing. This is the sum of the probabilities of the stock advancing on each exchange, weighted by the percentage of companies listed on each exchange. The formula is \(p_{\text{advance}} = p_{\text{exchange1}} \cdot p_{\text{advance | exchange1}} + p_{\text{exchange2}} \cdot p_{\text{advance | exchange2}}\). Substituting the given values, we get \(p_{\text{advance}} = 0.4446 \cdot 0.2648 + 0.5554 \cdot 0.2077 = 0.23308666\).
Step 3 :Next, use Bayes' theorem to calculate the conditional probability of the stock being listed on Exchange 1 given that it advanced. The formula is \(p_{\text{exchange1 | advance}} = \frac{p_{\text{advance | exchange1}} \cdot p_{\text{exchange1}}}{p_{\text{advance}}}\). Substituting the given values, we get \(p_{\text{exchange1 | advance}} = \frac{0.2648 \cdot 0.4446}{0.23308666} = 0.5050914539682365\).
Step 4 :Finally, round the result to four decimal places to get the final answer. The probability that the stock was listed on Exchange 1 given that it is advanced is \(\boxed{0.5051}\).