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Question 9, 8.6.25-BE
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On a recent date, a total of 8676 companies were listed with a certain stock exchange. On this day, the probabilities of advancing, declining, and unchanged company stocks appears in the table for the two exchanges. Find the probability the stock was listed on Exchange 1 given that it is advanced.
\begin{tabular}{l|c|c|c|c}
Exchange & \begin{tabular}{c}
Percentage of \\
Companies \\
Listed
\end{tabular} & \begin{tabular}{c}
Probability of \\
Stock \\
Advancing
\end{tabular} & \begin{tabular}{c}
Probability of \\
Stock \\
Declining
\end{tabular} & \begin{tabular}{c}
Probability of \\
Stock Remaining \\
Unchanged
\end{tabular} \\
\hline Exchange 1 & 0.4446 & 0.2648 & 0.6955 & 0.0397 \\
Exchange 2 & 0.5554 & 0.2077 & 0.7524 & 0.0399
\end{tabular}

The probability that the stock was listed on Exchange 1 given that it is advanced is $\square$.
(Round to four decimal places as needed.)
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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 \(\boxed{0.5051}\).