Coffee at Mom's Diner

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Alignments to Content Standards: S-CP.B.7

Task

At Momâ€™s diner, everyone drinks coffee. Let $C =$ the event that a randomly-selected customer puts cream in their coffee. Let $S =$ the event that a randomly-selected customer puts sugar in their coffee. Suppose that after years of collecting data, Mom has estimated the following probabilities:

$$ \begin{align} &P(C)=0.6 \\ &P(S)=0.5 \\ &P(C \text{ or } S)=0.7 \end{align} $$

Estimate $P(C \text{ and } S)$ and interpret this value in the context of the problem.

IM Commentary

This task assesses a student's ability to use the addition rule to compute a probability and to interpret a probability in context.

While the most obvious use of this task is as an assessment item, it could also be used in instruction as a proactice problem,

Solution

Using the addition rule, $P(C \text{ or } S) = P(C) + P(S) - P(C \text{ and } S)$, it follows that:

\begin{align} 0.7 &= 0.6 + 0.5 - P(C \text{ and } S) \\ P(C \text{ and } S) &= 0.6 + 0.5 - 0.7 \\ & = 0.4 \end{align}

The probability that a randomly-selected customer at Momâ€™s has both cream and sugar in his or her coffee is 0.4.