Sure! Let's solve the problem step-by-step.
To determine [tex]\( P(B \mid A) \)[/tex], which is the probability of event [tex]\( B \)[/tex] occurring given that event [tex]\( A \)[/tex] has occurred, we need to take into account the fact that the events [tex]\( A \)[/tex] (eating breakfast at a diner) and [tex]\( B \)[/tex] (watching cable) are independent.
In probability, when two events are independent, the occurrence of one event does not affect the occurrence of the other. Therefore, for independent events [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
[tex]\[ P(B \mid A) = P(B) \][/tex]
Here’s what we know:
- [tex]\( P(A) = 0.22 \)[/tex] (Probability of eating breakfast at a diner)
- [tex]\( P(B) = 0.46 \)[/tex] (Probability of watching cable)
Because the events are independent:
[tex]\[ P(B \mid A) = P(B) = 0.46 \][/tex]
So, the probability of watching cable given that someone ate breakfast at a diner is:
[tex]\[ P(B \mid A) = 0.46 \][/tex]
This is the final answer.
