The problem describes a scenario where a plane is missing, and it is presumed to have an equal probability of being in any of three regions. If the plane is actually down in region $i$, let $1 - \alpha_i$ denote the probability that the plane will be found upon a search of the $i$-th region, where $i = 1, 2, 3$. We are asked to find the conditional probability that the plane is in each region given that a search of that region was unsuccessful.

Let's define the following:

• $P(R_i)$: Probability that the plane is in region $i$. Since there is an equal probability of the plane being in any of the three regions, we have: $P(R_1) = P(R_2) = P(R_3) = \frac{1}{3}.$

• $P(\text{Unsuccessful Search in } R_i \mid R_i) = \alpha_i$: Probability that the search in region $i$ is unsuccessful given that the plane is in region $i$.

• $P(\text{Unsuccessful Search in } R_i)$: Probability that the search in region $i$ is unsuccessful.

Using Bayes' theorem, we want to find the conditional probability $P(R_i \mid \text{Unsuccessful Search in } R_i)$:

$P(R_i \mid \text{Unsuccessful Search in } R_i) = \frac{P(\text{Unsuccessful Search in } R_i \mid R_i) \cdot P(R_i)}{P(\text{Unsuccessful Search in } R_i)}.$

The probability $P(\text{Unsuccessful Search in } R_i)$ can be found using the law of total probability:

$P(\text{Unsuccessful Search in } R_i) = P(\text{Unsuccessful Search in } R_i \mid R_1) \cdot P(R_1) + P(\text{Unsuccessful Search in } R_i \mid R_2) \cdot P(R_2) + P(\text{Unsuccessful Search in } R_i \mid R_3) \cdot P(R_3).$

Substituting the values, we have:

$P(\text{Unsuccessful Search in } R_i) = \alpha_i \cdot \frac{1}{3} + \alpha_j \cdot \frac{1}{3} + \alpha_k \cdot \frac{1}{3},$ where $j$ and $k$ represent the other two regions (different from $i$).

Let's calculate the probabilities for parts (a), (b), and (c):

Part (a): Probability the plane is in Region 1 given Region 1 search was unsuccessful

$P(R_1 \mid \text{Unsuccessful in } R_1) = \frac{\alpha_1 \cdot \frac{1}{3}}{\alpha_1 \cdot \frac{1}{3} + \alpha_2 \cdot \frac{1}{3} + \alpha_3 \cdot \frac{1}{3}} = \frac{\alpha_1}{\alpha_1 + \alpha_2 + \alpha_3}.$

Part (b): Probability the plane is in Region 2 given Region 2 search was unsuccessful

$P(R_2 \mid \text{Unsuccessful in } R_2) = \frac{\alpha_2 \cdot \frac{1}{3}}{\alpha_1 \cdot \frac{1}{3} + \alpha_2 \cdot \frac{1}{3} + \alpha_3 \cdot \frac{1}{3}} = \frac{\alpha_2}{\alpha_1 + \alpha_2 + \alpha_3}.$

Part (c): Probability the plane is in Region 3 given Region 3 search was unsuccessful

$P(R_3 \mid \text{Unsuccessful in } R_3) = \frac{\alpha_3 \cdot \frac{1}{3}}{\alpha_1 \cdot \frac{1}{3} + \alpha_2 \cdot \frac{1}{3} + \alpha_3 \cdot \frac{1}{3}} = \frac{\alpha_3}{\alpha_1 + \alpha_2 + \alpha_3}.$

These results show that the conditional probabilities depend on the relative values of $\alpha_1$, $\alpha_2$, and $\alpha_3$.

Would you like further details or explanations on any part of this solution?

Here are five related questions to explore further:

1. How would the solution change if the probabilities of the plane being in each region were not equal?
2. What if the search success probabilities were different for each region?
3. How can Bayes' theorem be applied to more complex search and rescue scenarios?
4. What are some real-life applications of such probability models in search and rescue operations?
5. How do changes in the value of $\alpha_i$ affect the conditional probabilities calculated?

Tip: When dealing with conditional probability problems, always carefully define the given probabilities and use Bayes' theorem to relate them.