Let [tex]M,F,P[/tex] denote the events that a student is male, a student is female, and a student has a phone, respectively. Then we're given
[tex]P(M)=\dfrac{420}{800}[/tex]
[tex]P(F)=1-P(M)=\dfrac{380}{800}[/tex]
[tex]P(P)=\dfrac{325}{800}[/tex]
We want to determine [tex]P(M\cup P)[/tex], which is equivalent to
[tex]P(M\cup P)=P(M)+P(P)-P(M\cap P)[/tex]
By the law of total probability, we have
[tex]P(P)=P(M\cap P)+P(F\cap P)[/tex]
That is, any student belonging to [tex]P[/tex] will be either male or female, and the events [tex]M[/tex] and [tex]F[/tex] are mutually exclusive of one another (because they are complementary). We're told that
[tex]P(F\mid P)=\dfrac{200}{325}[/tex]
By definition of conditional probability,
[tex]P(F\mid P)=\dfrac{P(F\cap P)}{P(P)}=\dfrac{P(F\cap P)}{\frac{325}{800}}\implies P(F\cap P)=\dfrac{200}{800}[/tex]
and so
[tex]P(P)=P(M\cap P)+P(F\cap P)\iff\dfrac{325}{800}=P(M\cap P)+\dfrac{200}{800}\implies P(M\cap P)=\dfrac{125}{800}[/tex]
So we have
[tex]P(M\cup P)=\dfrac{420}{800}+\dfrac{325}{800}-\dfrac{125}{800}=\dfrac{620}{800}[/tex]
