Independent random variables are those variables of a random experiment whose values does not affect the value of the other variable. Two random variables $X$ and $Y$ are independent if, P$(X|Y)$ = $P(X)$, that is, the probability of $X$ if $Y$ has already occurred is not affected at all. The random variables $X$ and $Y$ are independent of each other iff,
$P(X$ and $Y)$ = $P(X)P(Y)$
Problem 1:
Aren tossed two dices, one was green in color and other one was black in color. Show that their outcomes are independent of each other.
Solution:
The sample space of throwing a dice = $\{1,\ 2,\ 3,\ 4,\ 5,\ 6\}$
The probability of getting any number in green dice = $\frac{1}{6}$
The probability of getting any number in black dice = $\frac{1}{6}$
The probability of getting any number in green and black dice = $\frac{1}{36}$
Hence, $P$(green and black) = $P$(green)$P$(black) and so these outcomes are independent random variables.