Dependent and independent events

Let's consider the experiment of throwing a dice, and consider the events A = "to extract $$6$$", B = "to extract an even number". It seems logical that if we knew that an even number had come out, then the probability that six was thrown was larger than what it would be if we did not have this information. Let's verify it:

We know that $$P(B) = \dfrac{1}{2}$$, by the rule of Laplace, and

$$$P(A\cap B)=\mbox{"probability of coming out a }6 \mbox{ and coming out an even number"}=$$$ $$$=\mbox{"probability of coming out a }6"=\dfrac{1}{6}$$$

$$$P(A/B)=\dfrac{P(A\cap B)}{P(B)}=\dfrac{\dfrac{1}{6}}{\dfrac{1}{2}}=\dfrac{1}{3}$$$

In particular, we have verified that our events $$A$$ and $$B$$ are dependent, since $$P(A/B)$$ is different from $$P(A)$$.

Carrying out a telephone poll, we have asked $$1000$$ persons if they believed it necessary to have more lighting in the street at night.

The poll was answered by $$480$$ men, of whom $$324$$ answered yes, and $$156$$ who said no, and $$520$$ women, of whom $$351$$ answered yes, and $$169$$ no. We wonder if men and women have a different opinion, or whether this is irrelevant to the question.

To see more clearly what they say, the best thing is to put the information in a table:

Let's consider the events $$A =$$"to want more light (to have answered yes)", $$B =$$"a man has answered".

We wonder if $$A$$ and $$B$$ are independent, that is to say, if the fact of wanting more light depends on whether one is a man or woman.

Let's calculate the probabilities:

$$$P(A)=\dfrac{324+351}{1000}=\dfrac{675}{1000}$$$ by the rule of Laplace (they are all those who have answered yes, adding up men and women).

$$$P(B)=\dfrac{480}{1000}$$$ the men who have answered us among all the calls.

$$$P(A\cap B)=\dfrac{324}{1000}$$$ those who are men and have answered yes.

It is satisfied that $$$\dfrac{324}{1000}=\dfrac{675}{1000}\cdot \dfrac{480}{1000}$$$ that is to say, that $$$P(A\cap B)=P(A)\cdot P(B)$$$ so the events are independent. In other words, the fact of being a man or a woman has not influenced whether one wants more light or not.