Please help me solve this statistics problem.
You are an owner of a firm that manages other small manufacturing companies, which pay you to employ temporary salespersons to sell their products. The owner of one of these firms wants to increase their sales and promised to give you a bonus of $40,000 if your temp salespeople sell at least 20 products tomorrow. Assume the cost of a single sales visit a temp sales reps cost you $200 and only a single visit per sales rep per day is allowed. Give that the probability of making a single sale is 0.25, determine the number of temporary salespersons you should plan to hire in order to maximize your profit.
To solve this problem, we need to determine the number of temporary salespersons we should plan to hire in order to maximize our profit.
Let's break down the problem and calculate the profit for each scenario:
1. If we hire no temporary salespersons, we have zero sales, and our profit is $0.
2. If we hire one temporary salesperson, the probability of making a sale is 0.25. If they make a sale, our profit will be ($40,000 - $200) = $39,800. However, if they don't make a sale, our profit will be -$200 (the cost of the sales visit). The expected profit can be calculated as 0.25 * $39,800 + 0.75 * -$200.
3. If we hire two temporary salespersons, the probability of making at least one sale increases. We can calculate the expected profit similarly to step 2, but this time taking into account two salespersons.
We can continue this process for different numbers of temporary salespersons and calculate the expected profit for each scenario.
Alternatively, we can use the concept of expected value to simplify this calculation. The expected value of a random variable is the average value it would take over a large number of repetitions. In our case, the random variable is the profit, and we want to maximize it.
Let's define the random variable X as the number of sales made by the temporary salespersons. The profit can be represented as a function of X:
Profit(X) = ($40,000 - $200X) if X ≥ 20, otherwise -$200X.
We know that the probability of making a sale is 0.25, so the number of sales X follows a binomial distribution with parameters n (number of temporary salespersons) and p (probability of success, which is 0.25).
To determine the optimal number of temporary salespersons, we need to find the value of n that maximizes the expected profit. We can use the following formula to calculate the expected profit (E(X)):
E(X) = ∑(x * P(X = x)), where x is the number of sales made and P(X = x) is the probability of making x sales.
We can calculate the expected profit for different values of n (number of temporary salespersons) and choose the value of n that gives us the highest expected profit.
I hope this explanation helps you understand the process of solving this statistics problem.