This problem involves conditional probability and can be solved using Bayes' Theorem. We are given information about a student's performance on multiple choice questions based on whether they work out the solution or guess the answer. We need to find the probability that the student guessed the answer, given that the answer obtained is correct.

Defining Events and Probabilities

Let's define the events involved:

• $\text{S}$: The event that the student works out the solution.
• $\text{G}$: The event that the student guesses the answer.
• $\text{C}$: The event that the answer is correct.
• $\text{W}$: The event that the answer is wrong.

We are given the following probabilities:

• Probability of working out the solution: $\text{P(S)} = 50\% = 0.50$.
• Probability of guessing the answer: Since these are the only two possibilities, $\text{P(G)} = 1 - \text{P(S)} = 1 - 0.50 = 0.50$.
• Probability of guessing correctly when guessing: $\text{P(C | G)} = 0.30$.
• Probability of working out incorrectly when working out: $\text{P(W | S)} = 0.10$.

From the probability of working out incorrectly, we can find the probability of working out correctly:

• Probability of working out correctly when working out: $\text{P(C | S)} = 1 - \text{P(W | S)} = 1 - 0.10 = 0.90$.

Applying Bayes' Theorem

We want to find the probability that the student guessed the answer, given that the answer is correct. This is the conditional probability $\text{P(G | C)}$.

Bayes' Theorem states:

\[ \text{P(G | C)} = \frac{\text{P(C | G)} \times \text{P(G)}}{\text{P(C)}} \]

To use this formula, we first need to calculate the total probability of getting a correct answer, $\text{P(C)}$. A correct answer can be obtained either by working out the solution correctly or by guessing correctly. These two events ($\text{S}$ and $\text{G}$) are mutually exclusive and cover all possibilities (they form a partition).

The probability of getting a correct answer is the sum of the probabilities of these two scenarios:

\[ \text{P(C)} = \text{P(C and S)} + \text{P(C and G)} \]

Using the formula for conditional probability ($\text{P(A and B)} = \text{P(A | B)} \times \text{P(B)}$ or $\text{P(A and B)} = \text{P(B | A)} \times \text{P(A)}$):

\[ \text{P(C and S)} = \text{P(C | S)} \times \text{P(S)} = 0.90 \times 0.50 = 0.45 \]

\[ \text{P(C and G)} = \text{P(C | G)} \times \text{P(G)} = 0.30 \times 0.50 = 0.15 \]

So, the total probability of a correct answer is:

\[ \text{P(C)} = 0.45 + 0.15 = 0.60 \]

Calculating the Required Probability

Now we have all the components to calculate $\text{P(G | C)}$ using Bayes' Theorem:

\[ \text{P(G | C)} = \frac{\text{P(C | G)} \times \text{P(G)}}{\text{P(C)}} = \frac{0.30 \times 0.50}{0.60} \]

\[ \text{P(G | C)} = \frac{0.15}{0.60} \]

To simplify the fraction $\frac{0.15}{0.60}$, we can multiply the numerator and denominator by 100:

\[ \frac{0.15 \times 100}{0.60 \times 100} = \frac{15}{60} \]

Simplifying the fraction $\frac{15}{60}$ by dividing both numerator and denominator by 15:

\[ \frac{15 \div 15}{60 \div 15} = \frac{1}{4} = 0.25 \]

Thus, the probability that the student guessed the answer, given that the answer is correct, is 0.25.

Summary of Probabilities

Using Bayes' Theorem to find the probability the student guessed, given the answer was correct:

\[ \text{P(G | C)} = \frac{\text{P(C and G)}}{\text{P(C)}} = \frac{0.15}{0.60} = 0.25 \]