TIFR ECE 2023 | Question: 8

Question

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):
(i) $P_{X_{1}}=P_{X_{10}}$
(ii) $P_{X_{1}, X_{10}}=P_{X_{10}, X_{1}}$
(iii) $P_{X_{1}, X_{2}, X_{3}}=P_{X_{3}, X_{7}, X_{5}}$

• A. Only (i)
• B. Only (i) and (ii)
• C. Only (i) and (iii)
• D. All of (i), (ii), and (iii)
• E. None of (i), (ii), or (iii)