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Complete the first row of the following table with the guaranteed utility the passengers will receive if they walk away with their original winnings, the new expected utility from taking the gamble, and whether they should do the video bonus question in this case.

[tex]\[
\begin{array}{|l|c|c|c|}
\hline
\text{Scenario} & \text{Guaranteed Utility (Utils)} & \text{Expected Utility (Utils)} & \text{Do the Video Bonus? (Yes or No)} \\
\hline
30\% \text{ chance of answering correctly} & \square & \square & \square \\
\$300 \text{ in winnings during the cab ride} & & & \\
\$200 \text{ initial vacation budget} & & & \\
\hline
\end{array}
\][/tex]

Now suppose that the chance that Felix, Janet, and Larry get the video bonus question right is again 60\%, but instead of winning \[tex]$100 during their ride, the family won \$[/tex]300. Taking this new information into account, complete the second row of the previous table.

Again, suppose that the chance the passengers get the video bonus question right is 60\%, and the total winnings from the cab ride return to \[tex]$100. However, suppose the initial vacation budget Felix, Janet, and Larry started with was only \$[/tex]200 instead of \$300. Based on this information, complete the final row of the previous table.

Sagot :

GinnyAnswer
Certainly! Here are the details filled into the table for each scenario:

\begin{tabular}{llll}
Scenario & \begin{tabular}{c}
Guaranteed Utility \\
(Utils)
\end{tabular} & \begin{tabular}{c}
Expected Utility \\
(Utils)
\end{tabular} & \begin{tabular}{c}
Do the Video Bonus? \\
(Yes or No)
\end{tabular} \\
\hline
[tex]$30 \%$[/tex] chance of answering correctly & [tex]$500$[/tex] & [tex]$530.0$[/tex] & Yes \\
[tex]$\$[/tex] 300[tex]$ in winnings during the cab ride & & & \\ $[/tex]\[tex]$ 200$[/tex] initial vacation budget & & & \\
\\
[tex]$60 \%$[/tex] chance of answering correctly & [tex]$500$[/tex] & [tex]$560.0$[/tex] & Yes \\
[tex]$\$[/tex] 300[tex]$ in winnings during the cab ride & & & \\ $[/tex]\[tex]$ 200$[/tex] initial vacation budget & & & \\
\\
[tex]$60 \%$[/tex] chance of answering correctly & [tex]$300$[/tex] & [tex]$360.0$[/tex] & Yes \\
[tex]$\$[/tex] 100[tex]$ in winnings during the cab ride \\ $[/tex]\[tex]$ 200$[/tex] initial vacation budget & & & \\
\end{tabular}


Explanation of each row:

1. First Row:
- Scenario: Passengers have a 30% chance of answering the video bonus question correctly, they won [tex]$300 during the cab ride, and they had an initial vacation budget of $[/tex]200.
- Guaranteed Utility: If they walk away with their winnings, the total amount of money they have is [tex]$300 (winnings) + $[/tex]200 (initial budget) = [tex]$500. - Expected Utility: Taking into account the 30% chance of winning an additional $[/tex]100 if they answer the video bonus question correctly, the expected utility is calculated to be [tex]$530.0. - Do the Video Bonus?: Since the expected utility ($[/tex]530.0) is higher than the guaranteed utility ([tex]$500), they should opt to do the video bonus. Hence, "Yes". 2. Second Row: - Scenario: The chance of answering correctly is 60%, they won $[/tex]300 during the cab ride, and they had an initial budget of [tex]$200. - Guaranteed Utility: The total money if they walk away is $[/tex]300 (winnings) + [tex]$200 (initial budget) = $[/tex]500.
- Expected Utility: With a 60% chance of answering correctly, the expected utility is [tex]$560.0. - Do the Video Bonus?: Since the expected utility ($[/tex]560.0) exceeds the guaranteed utility ([tex]$500), they should take the video bonus question. Hence, "Yes". 3. Third Row: - Scenario: The chance of answering correctly is 60%, they won $[/tex]100 during the cab ride, and they had an initial budget of [tex]$200. - Guaranteed Utility: If they walk away, they have $[/tex]100 (winnings) + [tex]$200 (initial budget) = $[/tex]300.
- Expected Utility: With a 60% chance of winning an additional [tex]$100, the expected utility is $[/tex]360.0.
- Do the Video Bonus?: Since the expected utility ([tex]$360.0) is greater than the guaranteed utility ($[/tex]300), they should take the video bonus. Hence, "Yes".
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