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Conditional probability helps us understand how likely an event is based on another event already occurring. In this oil exploration scenario, the geologists use conditional probability to evaluate the likelihood of different oil types being present given a specific soil type. To express this formally, we write the probability of finding soil given high-quality oil, medium-quality oil, or no oil as:

• \( P(\text{soil} | \text{high-quality oil}) = 0.20 \)
• \( P(\text{soil} | \text{medium-quality oil}) = 0.80 \)
• \( P(\text{soil} | \text{no oil}) = 0.20 \)

These probabilities suggest how soil types might relate to the likelihood of different oil quality findings. By understanding conditional probabilities, the company may make more data-driven decisions about where and how to drill. This understanding significantly updates their expectations based on new evidence from the soil test.

Before acquiring any new information, such as results from a soil test, prior probabilities represent the company's initial beliefs concerning each oil type. These probabilities suggest the state of knowledge regarding the land's potential. In this case, the initial (prior) probabilities assigned to finding each oil type in the land are:

• \( P(\text{high-quality oil}) = 0.50 \)
• \( P(\text{medium-quality oil}) = 0.20 \)
• \( P(\text{no oil}) = 0.30 \)

These numbers essentially convey the company's original expectations about the resource potential before obtaining additional evidence like soil tests. Since these beliefs are based on previous geologic information, they form the groundwork for any future probability assessments. To ascertain outcomes, the company will revise these prior probabilities using Bayes' Theorem once more data are available.

Revised probabilities incorporate new data to better predict an outcome; in oil drilling, they are crucial for strategic decisions. By applying Bayes' Theorem, the company can update prior probabilities using outcomes from the soil test:Bayes' Theorem states:\[ P(\text{A} | \text{B}) = \frac{P(\text{B} | \text{A}) \cdot P(\text{A})}{P(\text{B})} \]Here, \( P(\text{A} | \text{B}) \) denotes the revised probability of a specific oil type given the soil. The calculation includes:

• Sample likelihood: \( P(\text{soil}) = 0.32 \)
• Revised probability of high-quality: \( P(\text{high} | \text{soil}) = 0.3125 \)
• Revised probability of medium-quality: \( P(\text{medium} | \text{soil}) = 0.50 \)
• Revised probability of no oil: \( P(\text{none} | \text{soil}) = 0.1875 \)

These revised probabilities significantly influence whether and where further drilling should occur. For instance, a combined revised probability of finding some oil becomes \( P(\text{oil} | \text{soil}) = 0.8125 \). This informs the company that a higher chance exists to find oil than initially expected, guiding better investment choices and strategic planning.