Sick and not sick

Question

A sample of patients is classified according to 1) the gender, 2) the treatment received and 3) the response to treatment. The result of experiment is the following:

      sick          not sick

MALES

drug A

       200            300

drug B

       50             50 

FEMALES

drug A

       50            100

drug B

      200           370 

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Let $A=${the gender is male}$\rightarrow \mathbb{P}(A)=\frac{1}{2}$ and $A^c$={the gender is female}$\rightarrow \mathbb{P}(A^c)=1-\mathbb{P}(A)=\frac{1}{2}$. Let $H$={patient is treated with drug A} and $H^c$={patient is treated with drug B}. Then, let $G$={patient is sick} and $G^c$={patient is not sick}.

• 1) What is the probability to be sick after receiving the drug A for a male?

$\rightarrow \mathbb{P}(G|H\cap A)=\frac{\mathbb{P}(G\cap H\cap A)}{\mathbb{P}(H\cap A)}=\frac{\frac{200}{600}}{\mathbb{P}(A)\mathbb{P}(H|A)}=\frac{\frac{200}{600}}{\frac{1}{2}\cdot \frac{500}{600}}=0.799$

• 2) What is the probability to be sick after receiving the drug A for a female?

$\rightarrow \mathbb{P}(G|H\cap A^c)=\frac{\mathbb{P}(G\cap H\cap A^c)}{\mathbb{P}(H\cap A^c)}=\frac{\frac{50}{720}}{\mathbb{P}(A^c)\mathbb{P}(H|A^c)}=\frac{\frac{50}{720}}{\frac{1}{2}\cdot \frac{150}{720}}=0.666$

• 3) Considering the only males, do you recommend the drug A or B?

$\rightarrow \mathbb{P}(G^c\cap H|A)=\mathbb{P}(G^c|H)\mathbb{P}(H|A)=(1-\mathbb{P}(G|H))\mathbb{P}(H|A)=(1-\frac{200}{500})(\frac{500}{600})=0.5$

$\rightarrow \mathbb{P}(G^c\cap H^c|A)=\mathbb{P}(G^c|H^c)\mathbb{P}(H^c|A)=(1-\mathbb{P}(G|H^c))(1-\mathbb{P}(H|A))=(1-\frac{50}{100})(1-\frac{500}{600})=0.0833$

$\Rightarrow 0.5>0.0833\rightarrow $ drug A is recommend

• 4) Considering the only females, do you recommend the drug A or B?

$\rightarrow \mathbb{P}(G^c\cap H|A^c)=\mathbb{P}(G^c|H^c)\mathbb{P}(H|A^c)=(1-\mathbb{P}(G|H^c))(1-\mathbb{P}(H^c|A^c))=(1-\frac{200}{570})(\frac{570}{720})=0.514$

$\rightarrow \mathbb{P}(G^c\cap H^c|A^c)=\mathbb{P}(G^c|H)\mathbb{P}(H^c|A^c)=(1-\mathbb{P}(G|H))(1-\mathbb{P}(H|A^c))=(1-\frac{50}{150})(1-\frac{150}{720})=0.527$

$\Rightarrow 0.527>0.514\rightarrow $ drug B is recommend

Is it all correct?

If it is, point 5) asks if the conclusions make sense. What does it mean, in your opinion?