Question
A carnival is being setup in your hometown. From experience, the carnival organizer knows that the amount of money he will bring in depends on the weather.
If it is rainy, he will lose $1,000. If it is cloudy, he expects to earn $8,000. If it is sunny, he will bring in $9,000. The weather forecast shows that there is a 10% chance of rain, a 70% chance of sunny skies, and a 20% chance of clouds.
Show all of your calculations to find the expected value for this situation. Then interpret the expected value in context.
If it is rainy, he will lose $1,000. If it is cloudy, he expects to earn $8,000. If it is sunny, he will bring in $9,000. The weather forecast shows that there is a 10% chance of rain, a 70% chance of sunny skies, and a 20% chance of clouds.
Show all of your calculations to find the expected value for this situation. Then interpret the expected value in context.
Answers
GPT-4o mini
To find the expected value (EV) based on the potential financial outcomes and their probabilities, we can use the formula for expected value:
\[
EV = (P_1 \times X_1) + (P_2 \times X_2) + (P_3 \times X_3)
\]
Where:
- \(P_1, P_2, P_3\) are the probabilities of the respective outcomes.
- \(X_1, X_2, X_3\) are the amounts of money associated with the respective outcomes.
Let's identify the variables based on the weather conditions:
1. **Rainy**:
- Probability \(P_1 = 0.10\)
- Outcome \(X_1 = -1000\) (loss of $1,000)
2. **Cloudy**:
- Probability \(P_2 = 0.20\)
- Outcome \(X_2 = 8000\) (earn $8,000)
3. **Sunny**:
- Probability \(P_3 = 0.70\)
- Outcome \(X_3 = 9000\) (earn $9,000)
Now substituting these values into the expected value formula:
\[
EV = (0.10 \times -1000) + (0.20 \times 8000) + (0.70 \times 9000)
\]
Calculating each term:
1. **For Rainy**:
\[
0.10 \times -1000 = -100
\]
2. **For Cloudy**:
\[
0.20 \times 8000 = 1600
\]
3. **For Sunny**:
\[
0.70 \times 9000 = 6300
\]
Now, adding these values together to find the overall expected value:
\[
EV = -100 + 1600 + 6300
\]
\[
EV = 6800
\]
### Interpretation
The expected value (EV) of the carnival organizer's financial outcome, given the weather probabilities and their effects on earnings, is **$6,800**.
This means that, on average, considering the likelihood of each weather condition, the organizer can expect to make a profit of $6,800. This figure is a long-term average and does not imply that he will always earn this amount; it represents the expected outcome based on the probability distribution of the weather conditions. The organizer may use this information to make decisions, such as investing in better marketing or preparing for adverse weather events.
\[
EV = (P_1 \times X_1) + (P_2 \times X_2) + (P_3 \times X_3)
\]
Where:
- \(P_1, P_2, P_3\) are the probabilities of the respective outcomes.
- \(X_1, X_2, X_3\) are the amounts of money associated with the respective outcomes.
Let's identify the variables based on the weather conditions:
1. **Rainy**:
- Probability \(P_1 = 0.10\)
- Outcome \(X_1 = -1000\) (loss of $1,000)
2. **Cloudy**:
- Probability \(P_2 = 0.20\)
- Outcome \(X_2 = 8000\) (earn $8,000)
3. **Sunny**:
- Probability \(P_3 = 0.70\)
- Outcome \(X_3 = 9000\) (earn $9,000)
Now substituting these values into the expected value formula:
\[
EV = (0.10 \times -1000) + (0.20 \times 8000) + (0.70 \times 9000)
\]
Calculating each term:
1. **For Rainy**:
\[
0.10 \times -1000 = -100
\]
2. **For Cloudy**:
\[
0.20 \times 8000 = 1600
\]
3. **For Sunny**:
\[
0.70 \times 9000 = 6300
\]
Now, adding these values together to find the overall expected value:
\[
EV = -100 + 1600 + 6300
\]
\[
EV = 6800
\]
### Interpretation
The expected value (EV) of the carnival organizer's financial outcome, given the weather probabilities and their effects on earnings, is **$6,800**.
This means that, on average, considering the likelihood of each weather condition, the organizer can expect to make a profit of $6,800. This figure is a long-term average and does not imply that he will always earn this amount; it represents the expected outcome based on the probability distribution of the weather conditions. The organizer may use this information to make decisions, such as investing in better marketing or preparing for adverse weather events.