Let's address the questions in the image:

Problem Description

A stargazer believes a UFO crashed in one of three areas: Area 50, Area 51, or Area 52, with equal probability. The stargazer can only search one area. The probabilities of finding the object if it's in a specific area are:

• 59% in Area 50
• 66% in Area 51
• 56% in Area 52

Part (a): Probability of Finding the Object if It Crashed in Area 51

If the UFO crashed in Area 51, the probability that the stargazer will find it when searching in Area 51 is 66%, or 0.66.

Part (b): Probability that the Crash was in Area 51 Given an Unsuccessful Search in Area 51

We need to calculate the conditional probability that the UFO crashed in Area 51 given that the search in Area 51 was unsuccessful.

1. Prior Probabilities:

   • The probability of crashing in any area is 1/3, as the crash is equally likely in each area.

2. Probability of Not Finding the Object in Each Area:

   • Area 50: Probability of not finding the object = $1 - 0.59 = 0.41$
   • Area 51: Probability of not finding the object = $1 - 0.66 = 0.34$
   • Area 52: Probability of not finding the object = $1 - 0.56 = 0.44$

3. Total Probability of an Unsuccessful Search in Area 51: $P(\text{Unsuccessful search in 51}) = \left(\frac{1}{3} \times 0.41\right) + \left(\frac{1}{3} \times 0.34\right) + \left(\frac{1}{3} \times 0.44\right) = \frac{0.41 + 0.34 + 0.44}{3} = 0.3967$

4. Bayes' Theorem Application: $P(\text{Crashed in 51} | \text{Unsuccessful search in 51}) = \frac{P(\text{Unsuccessful search in 51} | \text{Crashed in 51}) \times P(\text{Crashed in 51})}{P(\text{Unsuccessful search in 51})}$ Substituting the values: $P(\text{Crashed in 51} | \text{Unsuccessful search in 51}) = \frac{0.34 \times \frac{1}{3}}{0.3967} = \frac{0.1133}{0.3967} \approx 0.2856$

Summary of Answers:

(a) The probability that the stargazer will find the object if it crashed in Area 51 is 0.66.

(b) The probability that the crash was in Area 51 given an unsuccessful search in Area 51 is approximately 0.2856.

Would you like more details on these calculations or have any questions? Here are some related questions to explore:

1. What is the probability of finding the object if the search is conducted in Area 52?
2. How would the probabilities change if the search success rates were different?
3. What is the total probability of finding the object in any of the areas with one search attempt?
4. Can the probabilities be calculated if more areas are added with different probabilities of finding the object?
5. What are some real-world applications of probability trees like this?

Tip: Conditional probability is often calculated using Bayes' theorem, which is helpful for updating probabilities based on new information.