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The state of being able to do something is known as probability. It's a branch of mathematics concerned with event randomness. The value is indicated from zero to one. The idea of probability was developed in mathematics to predict the possibility of events occurring. Probability is simply the probability of anything happening

The random variable of interest (total consumption of soft drinks) is

\({T_0} = {X_1} + {X_2} + \ldots + {X_{14}}\)

where each \({X_i},i = 1,2, \ldots ,14\)are random variable with normal distribution with given parameters (consumption of soft drinks on day i).

In order to compute the requested probability, mean value and standard deviation are required (to standardize \({T_0}\)and obtain standard normal distribution).

The mean value of the random variable \({T_0}\)is

\(\begin{aligned} \mu _{{T_0}}} &= E\left( {{X_1} + {X_2} + \ldots + {X_{14}}} \right &= E\left( {{X_1}} \right) + E\left( {{X_2}} \right) + \ldots + E\left( {{X_{14}}} \right)\\ &= 14 \cdot \mu \\ &= 14 \cdot 13\\ &= 182\end{aligned}\)

The variance of random variable \({T_0}\)is

\(\begin{aligned}\sigma _{{T_0}}^2 = V\left( {{T_0}} \right) &= V\left( {{X_1} + {X_2} + \ldots + {X_{14}}} \right) &= V\left( {{X_1}} \right) + V\left( {{X_2}} \right) + \ldots + V\left( {{X_{14}}} \right)\\ &= 14 \cdot {\sigma ^2}\\ &= 14 \cdot 4\\ &= 56\end{aligned}\)

\(\begin{array}{c}\sigma _{{T_0}}^2 = V\left( {{T_0}} \right) = V\left( {{X_1} + {X_2} + \ldots + {X_{14}}} \right) = V\left( {{X_1}} \right) + V\left( {{X_2}} \right) + \ldots + V\left( {{X_{14}}} \right)\\ = 14 \cdot {\sigma ^2}\\ = 14 \cdot 4\end{array}\)

The standard deviation of random variable \({T_0}\)is

\({\sigma _{{T_0}}} = \sqrt {56} = 7.483\)

The probability that some drinks are left at the end of \(14\) days, where total number of drinks is \(2 \cdot 6 \cdot 16 = 192\)(two six-packs, \(16\)bottles), is

\(\begin{array}{c}P\left( {{T_0} < 192} \right) = P\left( {\frac{{{T_0} - {\mu _{{T_0}}}}}{{{\sigma _{{T_0}}}}} < \frac{{192 - 182}}{{7.483}}} \right)\\ = P(Z < 1.34)\mathop = \limits^{(1)} 0.9099,\end{array}\)

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