Reasoning gone wrong?

I try to follow your approach (without explicit use of conditional probabilities) and give you the correct answer.

The test will be positive for $4/5$ of the total persons with the desease i.e. $\color{blue}{15}$ males and $\color{red}{4}$ females over $\color{blue}{30}+\color{red}{20}=50$, or for $1/5$ of the total persons without the desease i.e. $\color{blue}{15}$ males and $\color{red}{16}$ females over $50$. Hence the desired probability is $$\frac{\frac{4}{5}\cdot\frac{\color{blue}{15}}{50}+\frac{1}{5}\cdot\frac{\color{blue}{15}}{50}}{\frac{4}{5}\cdot\frac{\color{blue}{15}+\color{red}{4}}{50}+\frac{1}{5}\cdot\frac{\color{blue}{15}+\color{red}{16}}{50}}=\frac{75}{107}\approx 0.7009.$$

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Probability

Conditional Probability

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