Suppose a bag contains $5$ red balls, $3$ blue balls, and $2$ black balls. Balls are drawn without replacement until the bag is empty. Let $X_{i}$ be a random variable which takes value $1$ if the $i$-th ball drawn is red, value $2$ if that ball is blue, and $3$ if it is black. Let the joint probability mass function of the random variables be denoted by $P_{X_{1}, X_{2}, \ldots, X_{10}}$. Consider the following statements, where we write $f=g$ for two functions $f$ and $g$ defined on the same domain to mean that, for all possible elements in the domain, both functions map the same element to the same value (i.e., we write $f=g$ if $f(y)=g(y)$ for all $y$ in the domain):
- $P_{X_{1}}=P_{X_{10}}$
- $P_{X_{1}, X_{10}}=P_{X_{10}, X_{1}}$
- $P_{X_{1}, X_{2}, X_{3}}=P_{X_{3}, X_{7}, X_{5}}$
- Only $\text{(i)}$
- Only $\text{(i)}$ and $\text{(ii)}$
- Only $\text{(i)}$ and $\text{(iii)}$
- All of $\text{(i), (ii),}$ and $\text{(iii)}$
- None of $\text{(i), (ii),}$ or $\text{(iii)}$
