Problems from Dependent and independent events

Development:

We are considering two events, $C=$ "to be brunette", $O=$"to have brown eyes". For the statement, we know that $P(C)={\displaystyle \frac{6}{10}}=35$, $P(O)={\displaystyle \frac{7}{10}}$, $P(O\cup C)={\displaystyle \frac{8}{10}}={\displaystyle \frac{4}{5}}$.

They ask us about the probability of having brown eyes, knowing that the person is brunette, this is $P(O/C)$.

Applying the formula of the conditional probability $P(O/C)={\displaystyle \frac{P(O\cap C)}{P(C)}}$, yet we still do not know $P(O\cap C)$.

As we know the probability of the union, we can use the formula $P(O\cup C)=P(O)+P(C)-P(O\cap C)$.

By substituting, $${\displaystyle \frac{4}{5}}={\displaystyle \frac{7}{10}}+{\displaystyle \frac{3}{5}}-P(O\cap C)$$ and therefore, $$P(O\cap C)={\displaystyle \frac{7}{10}}+{\displaystyle \frac{3}{5}}-{\displaystyle \frac{4}{5}}={\displaystyle \frac{5}{10}}={\displaystyle \frac{1}{2}}$$

And so, $$P(O/C)={\displaystyle \frac{P(O\cap C)}{P(C)}}={\displaystyle \frac{{\displaystyle \frac{1}{2}}}{{\displaystyle \frac{3}{5}}}}={\displaystyle \frac{5}{6}}$$

To calculate if being a brunette is idenpendent of having brown eyes, we must ask ourselves whether $P(O\cap C)=P(O)\cdot P(C)$.

Substituting, ${\displaystyle \frac{1}{2}}\ne {\displaystyle \frac{7}{10}}\cdot {\displaystyle \frac{3}{5}}={\displaystyle \frac{21}{50}}$.

Therefore, two events are dependent.

Solution:

$P(O/C)={\displaystyle \frac{5}{6}}$. The two events are dependent.

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