To determine the probability that the stargazer will find the object given that it crashed in Area 51, we use the information and probabilities provided in the decision tree.

Information from the Problem:

1. The stargazer is equally unsure about which area the object crashed in. This means the initial probability of the crash being in each area is: $P(\text{Crashed in 50}) = P(\text{Crashed in 51}) = P(\text{Crashed in 52}) = \frac{1}{3}$

2. The probability of finding the object if the stargazer searches a specific area given it crashed there is:

   • $P(\text{Find} \mid \text{Crash in 50}) = 0.59$
   • $P(\text{Find} \mid \text{Crash in 51}) = 0.66$
   • $P(\text{Find} \mid \text{Crash in 52}) = 0.56$

Probability of Finding the Object Given It Crashed in Area 51:

We want to find $P(\text{Find} \mid \text{Crash in 51})$. According to the given probability:

$P(\text{Find} \mid \text{Crash in 51}) = 0.66$

This is because, if the object crashed in Area 51 and the stargazer searches Area 51, the probability of finding it there is 66%.

Thus, the probability that the stargazer will find the object if it crashed in Area 51 is 0.66 or 66%.

Would you like a deeper breakdown of how these conditional probabilities work, or do you have other questions about this scenario?

Follow-Up Questions:

1. What is the overall probability that the stargazer will find the object in any area they choose to search?
2. If the stargazer searches Area 50 and finds nothing, what is the probability the object crashed in Area 51?
3. How would the probability change if the stargazer searched Area 52 instead of Area 51?
4. What is the probability that the object crashed in Area 50 given that the stargazer searched and found it in Area 50?
5. How can the stargazer maximize their chances of finding the object, given the probabilities?

Tip:

To analyze such probability problems effectively, always use the conditional probability formulas and set up the events clearly to avoid confusion.