Part 1 of 2
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In a certain year, the probability that a stock was in the Information Technology sector was 0.1231 . The probability that the stock had a dividend yield of $2.00 \%$ or higher given that it was in the Information Technology sector was 0.2832 . The probability that a stock had a dividend yield of $2.00 \%$ or higher given that it was not in the Information Technology sector was 0.4346 . Find the probability that the stock was in the Information Technology sector given that it had a dividend yield of $2.00 \%$ or higher.

Let $\mathrm{F}$ be the event that a stock was in the Information Technology sector and $\mathrm{E}$ be the event that a stock had a dividend yield of $2.00 \%$ or higher.
\[
P(F)=\square \text { and } P\left(F^{\prime}\right)=\square
\]
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Step 1 ：Let \(F\) be the event that a stock was in the Information Technology sector and \(E\) be the event that a stock had a dividend yield of $2.00 \%$ or higher.

Step 2 ：We are given that \(P(F) = 0.1231\), \(P(E|F) = 0.2832\), and \(P(E|F') = 0.4346\).

Step 3 ：We can calculate \(P(E)\) as \(P(E) = P(E \cap F) + P(E \cap F')\), which is equal to \(P(E|F)P(F) + P(E|F')P(F')\).

Step 4 ：Substituting the given values, we get \(P(E) = 0.2832 \times 0.1231 + 0.4346 \times (1 - 0.1231)\).

Step 5 ：We are asked to find \(P(F|E)\), which can be calculated using Bayes' theorem: \(P(F|E) = \frac{P(E|F)P(F)}{P(E)}\).

Step 6 ：Substituting the given values and the calculated value of \(P(E)\) into the formula, we get \(P(F|E) = \frac{0.2832 \times 0.1231}{P(E)}\).

Step 7 ：Final Answer: The probability that the stock was in the Information Technology sector given that it had a dividend yield of $2.00 \%$ or higher is \(\boxed{0.0838}\).