Solutions - 5090-02

(1) Identify the reliability of each role:

Each staff member is available independently with probability $0.95$.

• Managers: need at least 1 of 2. $P=1-{(0.05)}^2$
• Cooks: need at least 1 of 3. $P=1-{(0.05)}^3$
• Servers: need all 3 of 3. $P={(0.95)}^3$
• Host: need 1 of 1. $P=0.95$

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(2) Multiply (independent):

$$\begin{array}{c}P[\text{enough staff}]=(1-{0.05}^2)(1-{0.05}^3)({0.95}^3)(0.95)\\ \\ \gg \gg 0.9975\cdot 0.999875\cdot 0.857375\cdot 0.95\gg \gg 0.8124\end{array}$$