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# Refer to a small consumer survey in which 10 people were asked to choose a cola among Coke, Pepsi, and RC. If each person chose a cola randomly, what is the probability that at least one person did not choose Coke?

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$\textbf{Multiplication principle: }$If one event can occur in $m$ ways AND a second event can occur in $n$ ways, then the number of ways that the two events can occur in sequence is then $m\cdot n$.

There are 3 options for each of the 10 people (Coke, Pepsi, and RC) and thus there are $3^{10}$ possible outcomes by the multiplication principle.

$\text{\# of possible outcomes}=3^{10}=59049$

The order in which we select the Coke answers does not matter (as a different order results in the same people answering Coke) and thus we need to use the definition of $\textbf{combination}$.

There are $C(10,0)$ ways to select 0 of the 10 people (who answer Coke), thus there are $59049-C(10,0)$ ways to select at least one person how answer Coke and thus there are $59049-C(10,0)$ favorable outcomes.

$\text{\# of favorable outcomes}=59049-C(10,0)=59049-1=59048$

The probability is the number of favorable outcomes divided by the number of possible outcomes:

\begin{align*} P(\text{At least 1 Coke})&=\dfrac{\text{\# of favorable outcomes}}{\text{\# of possible outcomes}} \\ &=\frac{59048}{59049} \\ &\approx 0.99998306 \\ &=99.998306\% \end{align*}

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