To find the probability that a randomly chosen person prefers either regular coffee or plain coffee, we will need to follow these steps:
1. Identify the probabilities:
- The probability that a person prefers regular coffee is the sum of the probabilities across all types of additives (plain, sugar, and creamer) for regular coffee.
So, [tex]\( P(\text{regular}) = 0.27 + 0.19 + 0.32 \)[/tex].
- The probability that a person prefers plain coffee is the sum of the probabilities for plain coffee across all types (regular and decaf).
So, [tex]\( P(\text{plain}) = 0.27 + 0.05 \)[/tex].
2. Account for overlap:
- The probability that a person prefers regular plain coffee is already included in both [tex]\( P(\text{regular}) \)[/tex] and [tex]\( P(\text{plain}) \)[/tex]. We must subtract this overlap to avoid double-counting.
So, [tex]\( P(\text{regular and plain}) = 0.27 \)[/tex].
3. Calculate [tex]\( P(\text{regular or plain}) \)[/tex]:
- Using the formula for the probability of the union of two events, we get:
[tex]\[
P(\text{regular or plain}) = P(\text{regular}) + P(\text{plain}) - P(\text{regular and plain})
\][/tex]
4. Substitute the probabilities:
- [tex]\( P(\text{regular}) = 0.27 + 0.19 + 0.32 = 0.78 \)[/tex]
- [tex]\( P(\text{plain}) = 0.27 + 0.05 = 0.32 \)[/tex]
- [tex]\( P(\text{regular and plain}) = 0.27 \)[/tex]
5. Perform the final calculation:
- [tex]\[
P(\text{regular or plain}) = 0.78 + 0.32 - 0.27 = 0.78
\][/tex]
Thus, the probability that a randomly chosen person prefers either regular coffee or plain coffee is [tex]\( \boxed{0.78} \)[/tex].
So, the correct answer is:
0.78
