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].
