The table shows coffee preferences from a survey.

\begin{tabular}{|l|l|l|l|}
\hline
Coffee Type & Plain & Sugar & Creamer \\
\hline
Regular & 0.27 & 0.19 & 0.32 \\
\hline
Decaf & 0.05 & 0.08 & 0.09 \\
\hline
\end{tabular}

If a person is chosen at random in this survey, what is the [tex]\( P(\text{decaf or sugar}) \)[/tex]?



Answer :

To determine the probability that a randomly selected person from the survey prefers either decaf coffee or coffee with sugar, we can use the principle of inclusion-exclusion from probability. Here's a detailed, step-by-step solution to the problem:

1. Identify the Individual Probabilities:

- First, we need to find the probability of selecting someone who prefers decaf coffee.

[tex]\( \text{P(decaf)} = \text{P(plain decaf)} + \text{P(sugar decaf)} + \text{P(creamer decaf)} \)[/tex]

Given the probabilities from the table:

[tex]\( \text{P(decaf)} = 0.05 + 0.08 + 0.09 = 0.22 \)[/tex]

- Next, determine the probability of selecting someone who prefers coffee with sugar.

[tex]\( \text{P(sugar)} = \text{P(sugar regular)} + \text{P(sugar decaf)} \)[/tex]

Given the probabilities from the table:

[tex]\( \text{P(sugar)} = 0.19 + 0.08 = 0.27 \)[/tex]

2. Determine the Probability of Both Events Occurring Together:

- In this context, the overlap between preferring decaf coffee and sugar coffee is the probability of someone preferring decaf sugar coffee.

[tex]\( \text{P(decaf and sugar)} = \text{P(sugar decaf)} \)[/tex]

From the table:

[tex]\( \text{P(decaf and sugar)} = 0.08 \)[/tex]

3. Use the Principle of Inclusion-Exclusion:

- To compute the probability of a person preferring either decaf coffee or coffee with sugar, we need to apply the principle of inclusion-exclusion:

[tex]\[ \text{P(decaf or sugar)} = \text{P(decaf)} + \text{P(sugar)} - \text{P(decaf and sugar)} \][/tex]

Substituting the values we have found:

[tex]\[ \text{P(decaf or sugar)} = 0.22 + 0.27 - 0.08 = 0.41 \][/tex]

Thus, the probability that a randomly selected person prefers either decaf coffee or coffee with sugar is [tex]\( \boxed{0.41} \)[/tex].