To compute the revised probabilities of a successful racquetball facility given favorable and unfavorable survey results, we can use Bayes' Theorem. Let's define the following events:

• $S$: Successful racquetball facility
• $U$: Unsuccessful racquetball facility
• $F$: Favorable survey result
• $N$: Unfavorable survey result

Given Data

• $P(S) = 0.3$ (prior probability of success)
• $P(U) = 1 - P(S) = 0.7$ (prior probability of failure)
• $P(F|S) = 0.8$ (probability of a favorable survey given success)
• $P(N|U) = 0.7$ (probability of an unfavorable survey given failure)
• Thus, $P(F|U) = 1 - P(N|U) = 0.3$ (probability of a favorable survey given failure)

Step 1: Calculate $P(F)$ (Total Probability of a Favorable Survey)

Using the law of total probability: $P(F) = P(F|S) \cdot P(S) + P(F|U) \cdot P(U)$ $P(F) = (0.8 \cdot 0.3) + (0.3 \cdot 0.7) = 0.24 + 0.21 = 0.45$

Step 2: Calculate $P(S|F)$ (Revised Probability of Success Given Favorable Survey)

Using Bayes' Theorem: $P(S|F) = \frac{P(F|S) \cdot P(S)}{P(F)}$ $P(S|F) = \frac{0.8 \cdot 0.3}{0.45} = \frac{0.24}{0.45} \approx 0.5333$

Step 3: Calculate $P(N)$ (Total Probability of an Unfavorable Survey)

Using the law of total probability: $P(N) = P(N|S) \cdot P(S) + P(N|U) \cdot P(U)$ $P(N) = (1 - P(F|S)) \cdot P(S) + P(N|U) \cdot P(U) = (0.2 \cdot 0.3) + (0.7 \cdot 0.7) = 0.06 + 0.49 = 0.55$

Step 4: Calculate $P(S|N)$ (Revised Probability of Success Given Unfavorable Survey)

Using Bayes' Theorem: $P(S|N) = \frac{P(N|S) \cdot P(S)}{P(N)}$ $P(S|N) = \frac{0.2 \cdot 0.3}{0.55} = \frac{0.06}{0.55} \approx 0.1091$

Summary of Results

• Revised probability of success given a favorable survey: $P(S|F) \approx 0.5333$ (53.33%)
• Revised probability of success given an unfavorable survey: $P(S|N) \approx 0.1091$ (10.91%)

Would you like more details or have any questions? Here are some related questions for further exploration:

1. What factors can influence the probabilities of success or failure for a racquetball facility?
2. How can Mark use these probabilities to make a decision about opening the facility?
3. What are other methods of market research that could complement the survey?
4. How might different marketing strategies impact the success rate?
5. What are the potential risks involved in opening a racquetball facility?

Tip: Always consider a range of scenarios and probabilities when making business decisions, as they can significantly impact your outcomes.