Problems from Dependent and independent events

In Barcelona, $$60\%$$ of the population are brunette, $$70\%$$ have brown eyes, and $$80\%$$ are brunette or have brown eyes.

We choose a person at random. If he or she is brunette: what is the probability to that he or she also has brown eyes? Is being a brunette idenpendent of having brown eyes, or is there a correlation?

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Development:

We are considering two events, $$C =$$ "to be brunette", $$O =$$"to have brown eyes". For the statement, we know that $$P(C)=\dfrac{6}{10}={3}{5}$$, $$P(O)=\dfrac{7}{10}$$, $$P(O\cup C)=\dfrac{8}{10}=\dfrac{4}{5}$$.

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 $$P(O/C)=\dfrac{P(O\cap C)}{P(C)}$$, 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, $$\dfrac{4}{5}=\dfrac{7}{10}+\dfrac{3}{5}-P(O\cap C)$$$and therefore, $$P(O\cap C)=\dfrac{7}{10}+\dfrac{3}{5}-\dfrac{4}{5}=\dfrac{5}{10}=\dfrac{1}{2}$$$

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

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, $$\dfrac{1}{2}\neq \dfrac{7}{10}\cdot\dfrac{3}{5}=\dfrac{21}{50}$$.

Therefore, two events are dependent.

Solution:

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

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