In a sample space there are two mutually exclusive events, $A _ { 1 }$ with probability 0.5 and $A _ { 2 }$ with probability 0.3, and $A _ { 3 } = \overline { A _ { 1 } \cup A _ { 2 } }$ is considered. Of a certain event $B$ with probability 0.4 it is known that it is independent of $A _ { 1 }$ and that the probability of the event $A _ { 3 } \cap B$ is 0.1 . With this data it is requested: a) ( 1 point) Calculate the probability of $A _ { 3 }$. b) ( 1.5 points) Decide whether $B$ and $A _ { 2 }$ are independent.
In a sample space there are two mutually exclusive events, $A _ { 1 }$ with probability 0.5 and $A _ { 2 }$ with probability 0.3, and $A _ { 3 } = \overline { A _ { 1 } \cup A _ { 2 } }$ is considered. Of a certain event $B$ with probability 0.4 it is known that it is independent of $A _ { 1 }$ and that the probability of the event $A _ { 3 } \cap B$ is 0.1 . With this data it is requested:
a) ( 1 point) Calculate the probability of $A _ { 3 }$.
b) ( 1.5 points) Decide whether $B$ and $A _ { 2 }$ are independent.