AUDIT DEADLINE

A consulting firm is preparing to provide financial advisory services for a client facing a critical audit deadline. The audit must be completed by the end of the week, and failure to meet the deadline could result in severe financial penalties for the client. If the audit is not concluded on time, the consulting firm has agreed to compensate the client with a penalty payment of $$51,000.$

The firm's manager is tasked with determining the size of the consulting team to ensure timely completion of the audit. Each consultant assigned to the project will be paid a flat fee of $$7,000$ for the week, and the available team sizes are $6,10,$ or $12$ consultants for Team A$,$ Team B$,$ and Team C respectively.

The manager estimates the probability of the $6-$person team meeting the audit deadline at $0\mathrm{.}19.$ For the $10-$person team, the probability of meeting the deadline is estimated at $0\mathrm{.}51,$ while the $12-$person team is expected to have a high probability of $0\mathrm{.}76$ for timely completion.

With the goal of minimizing expected costs, the manager wishes to determine the optimal team size required $,$ or $12).$ Use the provided decision tree to answer the following questions:

Payoff $($$$)$

Meet Deadline

$A_1$