The task involves finding the conditional probability that an animal is sick ($M$) given that the test is positive ($T$), written as $P(M \mid T)$. Using Bayes' theorem, this probability can be calculated as:

$P(M \mid T) = \frac{P(M \cap T)}{P(T)}$

Step 1: Compute $P(M \cap T)$

From the tree diagram: $P(M \cap T) = P(M) \cdot P(T \mid M)$ Substituting the values: $P(M) = 0.09, \quad P(T \mid M) = 0.92$ $P(M \cap T) = 0.09 \cdot 0.92 = 0.0828$

Step 2: Compute $P(\overline{M} \cap T)$

Similarly: $P(\overline{M} \cap T) = P(\overline{M}) \cdot P(T \mid \overline{M})$ Substituting the values: $P(\overline{M}) = 0.91, \quad P(T \mid \overline{M}) = 0.12$ $P(\overline{M} \cap T) = 0.91 \cdot 0.12 = 0.1092$

Step 3: Compute $P(T)$

The total probability of $T$ is: $P(T) = P(M \cap T) + P(\overline{M} \cap T)$ Substituting the values: $P(T) = 0.0828 + 0.1092 = 0.192$

Step 4: Compute $P(M \mid T)$

Now, apply Bayes' theorem: $P(M \mid T) = \frac{P(M \cap T)}{P(T)}$ Substituting the values: $P(M \mid T) = \frac{0.0828}{0.192} \approx 0.4313$

Final Answer:

The probability that an animal is sick given that the test is positive is approximately: $\boxed{0.4313}$

Would you like further details or have any questions?

Here are 5 related questions to expand your understanding:

1. What is the interpretation of conditional probability in medical testing scenarios?
2. How would the result change if the false positive rate ($P(T \mid \overline{M})$) were reduced?
3. How can Bayes' theorem help in evaluating the reliability of a diagnostic test?
4. What happens if the prevalence of the disease ($P(M)$) changes significantly?
5. How does the concept of sensitivity and specificity relate to the tree diagram probabilities?

Tip: Always calculate the total probability in Bayesian problems step-by-step to ensure precision, especially when combining multiple scenarios.