Don Carson and two colleagues are considering opening a law office in a large metropolitan area to make inexpensive legal services available to people who cannot otherwise afford these services. They intend to provide easy access for their clients by having the office open $360$ days per year, $16$ hours each day from $7$:$00$ a$.$m$.$ to $11$:$00$ p$.$m$.$ A lawyer, paralegal, legal secretary$,$ and clerk$-$receptionist would staff the office for each of the two $8-$hour shifts.

To determine the feasibility of the project, Don hired a marketing consultant to assist with market projections. The consultant$$s results show that if the firm spends $$506,000$ on advertising the first year, the number of new clients expected each day would have the following probability distribution:

Number of New Clients per Day Probability

$100\mathrm{.}20$

$200\mathrm{.}10$

$400\mathrm{.}30$

$600\mathrm{.}40$

Don and his associates believe these numbers to be reasonable and are prepared to spend the $$506,000$ on advertising. Other pertinent information about the operation of the office follows.

The only charge to each new client would be $$36$ for an initial consultation. The firm will accept on a contingency basis all cases that warrant further legal work, with the firm earning $30\%$ of any favorable settlements or judgments. Don estimates that $20\%$ of new client consultations will result in favorable settlements or judgments averaging $$15,300$ each. He does not expect repeat clients during the first year of operations.

The hourly wages for the staff are projected to be $$188$ for the lawyer, $$56$ for the paralegal, $$30$ for the legal secretary$,$ and $$20$ for the clerk$-$receptionist. Fringe benefits expense will be $40\%$ of the wages paid. A total of $400$ hours of overtime is expected for the year; this will be divided equally between the legal secretary and the clerk$-$receptionist positions. Overtime will be paid at one and one$-$half times the regular wage, and the fringe benefit expense will apply to the full wages.

Don has located $3,000$ square feet of suitable office space that rents for $$48$ per square foot annually. Associated expenses will be $$22,600$ for property insurance and $$32,600$ for utilities. The group must purchase malpractice insurance, expected to cost $$186,000$ annually.

The initial investment in office equipment will be $$60,600$; this equipment has an estimated useful life of four years. The cost of office supplies has been estimated to be $$16$ per expected new client consultation.

Required:

$1.$ Determine how many new clients must visit the law office that Don and his colleagues are considering for the venture to break even in the first year of operations. $($For purposes of this calculation, treat all labor costs as fixed with respect to number of new clients.$)$

$2.$ Using the probability information provided by the marketing consultant, determine whether it is feasible for the law office to achieve breakeven operations. Specifically, calculate the expected value of new clients for year $1,$ based on the probability data given above.