Context: this is a question from a textbook provided by my lecturer as practice. In the answer sheet, the answer is $0.22$, from my working, the answer is $0.27$. Since the lecturer has mentioned that some errors are present in the textbook I wanted to get a second opinion.
Question:
Probability of event A, normal supply of water from source A, is $0.7$.
Probability of event B, normal supply of water from source B, is $0.85$.
Probability of source B having below-normal supply of water given that source A is below normal is $0.3$.
Find the probability that only one of the two water sources will be below normal supply.
My working:
$P({\bar{B}}|{\bar{A}})$ = 0.3
$P({\bar{B}}|{\bar{A}}) = P({\bar{B}}{\bar{A}}) ÷ P({\bar{A}})$
$∴0.3 = P({\bar{B}}{\bar{A}}) ÷ (1 - 0.7)$
$P({\bar{B}}{\bar{A}}) = 0.3 * 0.3 = 0.09$
$P({\bar{B}}{\bar{A}}) + P(B) + P(A{\bar{B}}) = 1$
$0.09 + 0.85 + P(A{\bar{B}}) = 1$
$∴P(A{\bar{B}})=0.06$
$P(AB) = P(A) - P(A{\bar{B}})$
$= 0.7 - 0.06$
$= 0.64$
$P($only one of the two water sources will be below normal supply$)$
$= 1 - P[($normal supply from both sources$)∪($below normal supply from both sources$)]$
$=1-P[(AB)∪({\bar{A}}{\bar{B}})]$
Since $AB$ and ${\bar{A}}{\bar{B}}$ are mutually exclusive events,
$=1-[P(AB)+P({\bar{A}}{\bar{B}})]$
$=1-P(AB)-P({\bar{A}}{\bar{B}})$
$=1-0.64-0.09$
$= 0.27$
Did I make a logical fallacy or something that would lead to the difference between the given answer and my answer? No textbook solution provided.