Problem
Solve the problem. A lab orders a shipment of frogs each week. Prices for the weekly shipments of frogs follow the distribution below: \begin{tabular}{|c|c|c|c|} \hline Price & $\$ 9.00$ & $\$$ & $\$$ \\ & & 11.00 & 15.00 \\ \hline Probability & 0.15 & 0.45 & 0.40 \\ \hline \end{tabular} Using this probability model to create a budget for the future, how much should the lab expect the shipments to cost per week?
Solution
Step 1 ：Calculate the expected cost per week using the formula: \(E = x_1 \cdot p_1 + x_2 \cdot p_2 + x_3 \cdot p_3\)
Step 2 ：Substitute the given values: \(E = 9.00 \cdot 0.15 + 11.00 \cdot 0.45 + 15.00 \cdot 0.40\)
Step 3 ：Calculate the expression: \(E = 1.35 + 4.95 + 6.00\)
Step 4 ：Simplify the expression: \(E = 12.30\)
Step 5 ：Therefore, the lab should expect the shipments to cost \(\boxed{12.30}\) per week.
