Consider a experiment that you ask the first person you meet on the street in which month her birthday falls.
The sample space is
$S= \{ Janua\color{red}ry , Feb\color{red}rua\color{red}ry ,Ma\color{red}rch ,Ap\color{red}ril ,May, June ,July, August ,Septembe\color{red}r ,Octobe\color{red}r, Novembe\color{red}r, Decembe\color{red}r \}$
Now consider the events:
L : "born in a long month i.e months with 31 days"
R: "born in a month with the letter r"
H : "born in the first half of the year"
Clearly,
$P(H)= 1/2$
$P(H/R)= 1/2$
So $H$ and $R$ are independent
Also
$P(H \cap L ) =1/4$ and $P(L)=7/12$
but , $1/2×7/12 \neq 1/4$
So we conclude $L$ and $H$ are dependent.
Why it is that $R$ & $H$ are independent but $L$ & $H$ are dependent?
Till now i have considered that physical dependence and independence ( means what we refer even non mathematically) was same as stochastic dependence and independence.
But after this above e.g, i am not able to actually understand what does stochastic independence means?
So to be really be sure that events are stochastically independence we have to rely on mathematical formulas (not that i am complaining) ?