Answer :
Certainly! To determine the probability that a randomly selected resident opposes spending money on plumbing issues and favors spending money on walkways, we will need to apply the concept of conditional probability.
Firstly, let's define the two given probabilities:
- The probability that a resident opposes spending money on plumbing issues ([tex]\( P(A) \)[/tex]) is [tex]\( 0.8 \)[/tex].
- The probability that a resident favors spending money on improving walkways given that the resident opposes spending money on plumbing issues ([tex]\( P(B|A) \)[/tex]) is [tex]\( 0.85 \)[/tex].
We need to determine the probability that a resident both opposes spending money on plumbing issues and favors spending money on walkways ([tex]\( P(A \cap B) \)[/tex]).
According to the definition of conditional probability, we have:
[tex]\[ P(B|A) = \frac{P(A \cap B)}{P(A)} \][/tex]
Rearranging this formula to solve for [tex]\( P(A \cap B) \)[/tex], we get:
[tex]\[ P(A \cap B) = P(A) \times P(B|A) \][/tex]
Now, substituting the given probabilities into this equation, we calculate:
[tex]\[ P(A \cap B) = 0.8 \times 0.85 \][/tex]
Let's perform the multiplication:
[tex]\[ P(A \cap B) = 0.8 \times 0.85 = 0.68 \][/tex]
Therefore, the probability that a randomly selected resident opposes spending money on plumbing issues and favors spending money on walkways is [tex]\( 0.68 \)[/tex].
Firstly, let's define the two given probabilities:
- The probability that a resident opposes spending money on plumbing issues ([tex]\( P(A) \)[/tex]) is [tex]\( 0.8 \)[/tex].
- The probability that a resident favors spending money on improving walkways given that the resident opposes spending money on plumbing issues ([tex]\( P(B|A) \)[/tex]) is [tex]\( 0.85 \)[/tex].
We need to determine the probability that a resident both opposes spending money on plumbing issues and favors spending money on walkways ([tex]\( P(A \cap B) \)[/tex]).
According to the definition of conditional probability, we have:
[tex]\[ P(B|A) = \frac{P(A \cap B)}{P(A)} \][/tex]
Rearranging this formula to solve for [tex]\( P(A \cap B) \)[/tex], we get:
[tex]\[ P(A \cap B) = P(A) \times P(B|A) \][/tex]
Now, substituting the given probabilities into this equation, we calculate:
[tex]\[ P(A \cap B) = 0.8 \times 0.85 \][/tex]
Let's perform the multiplication:
[tex]\[ P(A \cap B) = 0.8 \times 0.85 = 0.68 \][/tex]
Therefore, the probability that a randomly selected resident opposes spending money on plumbing issues and favors spending money on walkways is [tex]\( 0.68 \)[/tex].
