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In everyday speech, the word probability has numerous meanings. Two of them are particularly crucial for the advancement and application of probability theory in mathematics. One is the interpretation of probabilities as relative frequencies, which may be demonstrated using basic games such as coins, cards, dice, and roulette wheels.

The expected value of the volume

\({V_o} = 27{X_1} + 125{X_2} + 512{X_3}\)

is

\(\begin{array}{c}E\left( {{V_o}} \right) = E\left( {27{X_1} + 125{X_2} + 512{X_3}} \right)\\\mathop = \limits^{(1)} 27E\left( {{X_1}} \right) + 125E\left( {{X_2}} \right) + 512E\left( {{X_3}} \right)\\\mathop = \limits^{(2)} 27 \cdot 200 + 125 \cdot 250 + 512 \cdot 100\\ = 87,850\end{array}\)

(1): this stands for any random variables,

(2): the expectations are given in the exercise.

The variance of the volume is

\(\begin{array}{c}V\left( {{V_o}} \right) = V\left( {27{X_1} + 125{X_2} + 512{X_3}} \right)\\\mathop = \limits^{(3)} {27^2}V\left( {{X_1}} \right) + {125^2}V\left( {{X_2}} \right) + {512^2}V\left( {{X_3}} \right)\\\mathop = \limits^{(4)} {27^2} \cdot {10^2} + {125^2} \cdot {12^2} + {512^2} \cdot {8^2}\\ = 19,100,116\end{array}\)

(3): the given random variables are independent therefore this equality stands,

(4): the standard variations of the random variables are given in the exercise.

The standard deviation of the volume is

\({\sigma _{{V_o}}} = \sqrt {19,100,116} = 4,370.37\)

Therefore, the probability that the total volume shipped is at most \(100,000{\rm{f}}{{\rm{t}}^3}\)can be computed by standardizing random variable\({V_o}\). The following holds

\(\begin{array}{c}P\left( {{V_o} \le 100,000} \right) = P\left( {\frac{{{V_o} - E\left( {{V_o}} \right)}}{{{\sigma _{{V_o}}}}} \le \frac{{100,000 - 87,850}}{{4,370.37}}} \right)\\ = P(Z \le 2.78)\mathop = \limits^{(5)} 0.9973\end{array}\)

(5): from the normal probability table in the appendix. The probability can also be computed with a software.