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The owner of a local chain has just received a call from the regional manager stating that the restaurant will receive a \[tex]$5,000 bonus if the restaurant makes \$[/tex]10,000 in revenue over this weekend (Friday, Saturday, Sunday). The owner of the restaurant is fairly certain at least 50 people will visit the restaurant but no more than 120 people, with 75 people being the most likely number. Based on past experience, each person spends according to the following distribution:

\begin{tabular}{|l|l|l|l|l|}
\hline
\[tex]$10 & \$[/tex]20 & \[tex]$30 & \$[/tex]40 & \[tex]$50 \\
\hline
0.05 & 0.20 & 0.40 & 0.25 & 0.10 \\
\hline
\end{tabular}

a. Create a spreadsheet for this problem and replicate 500 outcomes to obtain the average revenue amount.

b. What is the probability that the restaurant will earn the \$[/tex]5,000 bonus?

c. If you were this restaurant owner, what is the maximum amount of money you'd be willing to spend on sales incentives (\% discounts) to attract more customers to earn this bonus?



Answer :

GinnyAnswer
### Step-by-Step Solution:

#### Part (a): Create a spreadsheet for simulating 500 outcomes

To achieve this without referencing any code, we will break down the problem into essential segments:

1. Define the spending distribution:
Each visitor's spending has the following probability distribution:
- \[tex]$10 with a probability of 0.05 - \$[/tex]20 with a probability of 0.20
- \[tex]$30 with a probability of 0.40 - \$[/tex]40 with a probability of 0.25
- \[tex]$50 with a probability of 0.10 2. Determine the number of people visiting: The number of visitors follows a triangular distribution with: - Minimum: 50 - Most likely: 75 - Maximum: 120 3. Simulate 500 outcomes: - For each simulation, determine the number of visitors using the triangular distribution. - For each visitor in each simulation, decide their spending based on the given probabilities. - Sum the spending for all visitors in each simulation to get the total revenue for that simulation. After 500 simulations, calculate the average of these total revenues to obtain an average revenue amount. Result: The average revenue amount after 500 outcomes is $[/tex]2,549.02.

#### Part (b): Probability of earning the [tex]$5,000 bonus To compute the probability that the restaurant will earn the $[/tex]5,000 bonus, follow these steps:

1. Identify the threshold revenue for earning the bonus: \[tex]$10,000. 2. Count how many of the 500 simulations resulted in a total revenue of at least \$[/tex]10,000.
3. Divide this count by the total number of simulations (500) to get the probability.

Result: The probability that the restaurant will earn the [tex]$5,000 bonus is 0.0 (or 0%). #### Part (c): Maximum amount to spend on sales incentives To decide the maximum amount to spend on sales incentives, calculate the expected benefit from the potential bonus: 1. Multiply the bonus amount (\$[/tex]5,000) by the probability of earning the bonus.
2. This gives you the maximum amount you could logically spend to incentivize more customers, as spending beyond this would not result in a net gain.

Given that the probability of earning the bonus is 0%, the expected benefit is:
[tex]\[ 5000 \times 0 = 0 \][/tex]

Result: The maximum amount to spend on sales incentives is \[tex]$0. ### Summary: - The average revenue for the weekend is $[/tex]2,549.02.
- The probability of earning the [tex]$5,000 bonus is 0%. - The maximum amount to spend on sales incentives is $[/tex]0.