To determine if the local club should expect to earn a profit and assess the risk of the investment, we need to analyze the expected profit or loss based on the given probabilities.
Calculating Expected Profit:
- Investment: $10,000
- Revenue if no rain: $15,000
- Probability of rain: 30% (0.3)
- Probability of no rain: 70% (0.7)
1. Calculate the Expected Profit:
- Profit if it doesn't rain: Revenue - Investment = $15,000 - $10,000 = $5,000
- Loss if it rains: Investment = $10,000
Next, we calculate the expected profit based on these scenarios:
\[ \text{Expected Profit} = P(\text{No Rain}) \times \text{Profit if No Rain} + P(\text{Rain}) \times \text{Loss if Rain} \]
Substituting the values:
\[ \text{Expected Profit} = (0.7 \times 5000) + (0.3 \times (-10000)) \] \[ = (0.7 \times 5000) + (0.3 \times -10000) \] \[ = 3500 - 3000 \] \[ = 500 \]
Conclusion on Profitability:
- The expected profit is $500, which is greater than $0, so the club would expect to earn a profit on average from this investment.
Risk Assessment:
To determine whether the investment is risky, we consider the condition that if the return is less than $8,000, the investment is classified as risky.
- The maximum potential return (if no rain occurs) is $15,000.
- The expected profit (which is less than $8,000 if we consider the risk of losing all when it rains) doesn't trigger a higher guarantee.
In this case, since we're looking at the expected profit based on probabilities and the potential loss involved (which may lead us to consider the downside risk), the club’s investment shows potential profit, but the presence of a 30% chance of complete loss definitely indicates a risk element.
Conclusion on Risk:
Yes, while the investment does provide an expected profit, it comes with substantial risk due to the rain, which can lead to a total loss of the investment. Thus, despite an expectation of profit, the investment is indeed risky.
Z-score Formula Denominator:
While the z-score calculation typically uses standard deviation and means for a normal distribution, it seems you may want to compare this profit and risk to some average or standard performance metrics.
The denominator of the z-score formula is:
\[ z = \frac{(X - \mu)}{\sigma} \]
Where:
- \(X\) = the observed value
- \(\mu\) = mean of the distribution
- \(\sigma\) = standard deviation of the distribution
However, the specific context or values for \(\mu\) and \(\sigma\) haven't been provided to apply it to your scenario further. If you need specific values to compute the z-score, please provide those data points.