Sure! Let’s work through this step by step: --- ## **a. Probability of Finding Oil (to 2 decimals)** The probability of finding **any oil** is the sum of the probabilities of high-quality and medium-quality oil: \[ P(\text{oil}) = P(\text{high-quality oil}) + P(\text{medium-quality oil}) \] \[ P(\text{oil}) = .45 + .25 = .70 \] **Answer:** `.70` --- ## **b. Bayes’ Theorem for Revised Probabilities** ### **Given:** - \(P(\text{high-quality oil}) = .45\) - \(P(\text{medium-quality oil}) = .25\) - \(P(\text{no oil}) = .30\) - \(P(\text{soil}|\text{high-quality oil}) = .15\) - \(P(\text{soil}|\text{medium-quality oil}) = .85\) - \(P(\text{soil}|\text{no oil}) = .15\) #### **Step 1: Find \(P(\text{soil})\) using the Law of Total Probability** \[ P(\text{soil}) = P(\text{soil}|\text{high-quality oil})P(\text{high-quality oil}) + P(\text{soil}|\text{medium-quality oil})P(\text{medium-quality oil}) + P(\text{soil}|\text{no oil})P(\text{no oil}) \] Plug in the values: \[ P(\text{soil}) = .15 \times .45 + .85 \times .25 + .15 \times .30 \] \[ = .0675 + .2125 + .045 = .325 \] --- #### **Step 2: Apply Bayes’ Theorem** \[ P(A|B) = \frac{P(B|A)P(A)}{P(B)} \] ##### **1. \(P(\text{high-quality oil}|\text{soil})\):** \[ P(\text{high-quality oil}|\text{soil}) = \frac{P(\text{soil}|\text{high-quality oil}) \cdot P(\text{high-quality oil})}{P(\text{soil})} = \frac{.15 \times .45}{.325} = \frac{.0675}{.325} \approx .2077 \] ##### **2. \(P(\text{medium-quality oil}|\text{soil})\):** \[ P(\text{medium-quality oil}|\text{soil}) = \frac{.85 \times .25}{.325} = \frac{.2125}{.325} \approx .6538 \] ##### **3. \(P(\text{no oil}|\text{soil})\):** \[ P(\text{no oil}|\text{soil}) = \frac{.15 \times .30}{.325} = \frac{.045}{.325} \approx .1385 \] --- #### **Step 3: New Probability of Finding Oil** \[ P(\text{oil}|\text{soil}) = P(\text{high-quality oil}|\text{soil}) + P(\text{medium-quality oil}|\text{soil}) \] \[ = .2077 + .6538 = .8615 \] --- ### **Step 4: Most Likely Quality of Oil** - Highest probability after revision: **Medium-quality oil** (\(.6538\)) --- ## **Summary Table** | Probability | Value | |----------------------------------------|----------| | \(P(\text{high-quality oil}|\text{soil})\) | .2077 | | \(P(\text{medium-quality oil}|\text{soil})\) | .6538 | | \(P(\text{no oil}|\text{soil})\) | .1385 | | \(P(\text{oil}|\text{soil})\) | .8615 | --- ### **Answers for the Blanks:** - **a.** `.70` - **P(high-quality oil|soil):** `.2077` - **P(medium-quality oil|soil):** `.6538` - **P(no oil|soil):** `.1385` - **New probability of finding oil:** `.8615` - **Most likely oil quality:** `Medium-quality oil` Let me know if you want any step explained further!

Sure! Here’s the complete solution without using a table format. --- ## **a. Probability of Finding Oil (to 2 decimals)** To find the probability of discovering any type of oil, we add the probabilities of high-quality and medium-quality oil: \[ P(\text{oil}) = P(\text{high-quality oil}) + P(\text{medium-quality oil}) = 0.45 + 0.25 = 0.70 \] **Answer:** `.70` --- ## **b. Bayes’ Theorem for Revised Probabilities** ### **Given Probabilities:** - \(P(\text{high-quality oil}) = 0.45\) - \(P(\text{medium-quality oil}) = 0.25\) - \(P(\text{no oil}) = 0.30\) - \(P(\text{soil}|\text{high-quality oil}) = 0.15\) - \(P(\text{soil}|\text{medium-quality oil}) = 0.85\) - \(P(\text{soil}|\text{no oil}) = 0.15\) ### **Step 1: Calculate \(P(\text{soil})\)** Using the Law of Total Probability: \[ P(\text{soil}) = P(\text{soil}|\text{high-quality oil}) \cdot P(\text{high-quality oil}) + P(\text{soil}|\text{medium-quality oil}) \cdot P(\text{medium-quality oil}) + P(\text{soil}|\text{no oil}) \cdot P(\text{no oil}) \] Calculating each term: \[ P(\text{soil}) = 0.15 \times 0.45 + 0.85 \times 0.25 + 0.15 \times 0.30 \] \[ = 0.0675 + 0.2125 + 0.045 = 0.325 \] --- ### **Step 2: Apply Bayes’ Theorem** Using Bayes' theorem: \[ P(A|B) = \frac{P(B|A)P(A)}{P(B)} \] #### **1. Calculate \(P(\text{high-quality oil}|\text{soil})\)** \[ P(\text{high-quality oil}|\text{soil}) = \frac{P(\text{soil}|\text{high-quality oil}) \cdot P(\text{high-quality oil})}{P(\text{soil})} \] \[ = \frac{0.15 \times 0.45}{0.325} = \frac{0.0675}{0.325} \approx 0.2077 \] #### **2. Calculate \(P(\text{medium-quality oil}|\text{soil})\)** \[ P(\text{medium-quality oil}|\text{soil}) = \frac{P(\text{soil}|\text{medium-quality oil}) \cdot P(\text{medium-quality oil})}{P(\text{soil})} \] \[ = \frac{0.85 \times 0.25}{0.325} = \frac{0.2125}{0.325} \approx 0.6538 \] #### **3. Calculate \(P(\text{no oil}|\text{soil})\)** \[ P(\text{no oil}|\text{soil}) = \frac{P(\text{soil}|\text{no oil}) \cdot P(\text{no oil})}{P(\text{soil})} \] \[ = \frac{0.15 \times 0.30}{0.325} = \frac{0.045}{0.325} \approx 0.1385 \] --- ### **Step 3: Calculate New Probability of Finding Oil** The new probability of finding any type of oil is: \[ P(\text{oil}|\text{soil}) = P(\text{high-quality oil}|\text{soil}) + P(\text{medium-quality oil}|\text{soil}) \] \[ = 0.2077 + 0.6538 = 0.8615 \] --- ### **Step 4: Identify Most Likely Quality of Oil** The quality of oil that is most likely to be found is determined by the highest probability among the revised probabilities: - \(P(\text{high-quality oil}|\text{soil}) \approx 0.2077\) - \(P(\text{medium-quality oil}|\text{soil}) \approx 0.6538\) - \(P(\text{no oil}|\text{soil}) \approx 0.1385\) **Most Likely Quality:** `Medium-quality oil` --- ## **Final Answers** - **a. Probability of finding oil:** `.70` - **Revised probabilities:** - \(P(\text{high-quality oil}|\text{soil})\): `.2077` - \(P(\text{medium-quality oil}|\text{soil})\): `.6538` - \(P(\text{no oil}|\text{soil})\): `.1385` - **New probability of finding oil:** `.8615` - **Most likely quality of oil:** `Medium-quality oil` If you have any further questions, feel free to ask!