Nine out of 10 calls that Juan makes to his wife at 4:00 in the afternoon, she is talking to her mother, because the line sounds busy, if there is a call at 4:00, what is the probability that the line sound busy?

STEP BY STEP SOLUTION: Let's denote events as follows: - Event A: Juan makes a call at 4:00. - Event B: The line sounds busy. - Event C: Juan's wife is talking to her mother. We are interested in finding the conditional probability $$ \(P(B|A)\)$$, which represents the probability that the line sounds busy given that Juan makes a call at 4:00. Using conditional probability formula: $$ \[P(B|A) = \frac{P(B \cap A)}{P(A)}\] $$ From the information provided, we know that $$ \(P(A) = 1\) $$ (since Juan makes a call at 4:00). We also know that out of 10 calls, 9 times the line is busy while his wife is talking to her mother. So, $$ \(P(B \cap A) = \frac{9}{10}\) $$. Plug these values into the formula: $$ \[P(B|A) = \frac{\frac{9}{10}}{1} = \frac{9}{10}\] $$ Therefore, the probability that the line sounds busy, given that Juan makes a call at 4:00, is $$ \(\frac{9}{10}\) $$, or 90%.