$A$ bag contains $2$ white and $1$ red balls. One ball is drawn at random and then put back in the box after noting its colour. The process is repeated again. If $X$ denotes the number of red balls recorded in the two draws,describe $X$.
(N/A) Let the balls in the bag be denoted by $w_{1}, w_{2}, r$. The sample space $S$ for two draws with replacement is given by:
$S = \{w_{1}w_{1}, w_{1}w_{2}, w_{2}w_{1}, w_{2}w_{2}, w_{1}r, w_{2}r, rw_{1}, rw_{2}, rr\}$
Here,$X$ is a random variable representing the number of red balls in the two draws.
For outcomes where no red ball is drawn: $X(w_{1}w_{1}) = X(w_{1}w_{2}) = X(w_{2}w_{1}) = X(w_{2}w_{2}) = 0$.
For outcomes where exactly one red ball is drawn: $X(w_{1}r) = X(w_{2}r) = X(rw_{1}) = X(rw_{2}) = 1$.
For the outcome where two red balls are drawn: $X(rr) = 2$.
Thus,$X$ is a random variable that can take values $0, 1,$ or $2$.
$S = \{w_{1}w_{1}, w_{1}w_{2}, w_{2}w_{1}, w_{2}w_{2}, w_{1}r, w_{2}r, rw_{1}, rw_{2}, rr\}$
Here,$X$ is a random variable representing the number of red balls in the two draws.
For outcomes where no red ball is drawn: $X(w_{1}w_{1}) = X(w_{1}w_{2}) = X(w_{2}w_{1}) = X(w_{2}w_{2}) = 0$.
For outcomes where exactly one red ball is drawn: $X(w_{1}r) = X(w_{2}r) = X(rw_{1}) = X(rw_{2}) = 1$.
For the outcome where two red balls are drawn: $X(rr) = 2$.
Thus,$X$ is a random variable that can take values $0, 1,$ or $2$.
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