2025 TMJC P2 Q10

Wardrobe \(A\) contains \(4\) distinct blouses, labelled \(T_1, T_2, T_3\) and \(T_4\). Wardrobe \(B\) contains \(3\) distinct pairs of pants, labelled \(P_1, P_2\) and \(P_3\). The following table shows the formality (formal or casual) of all the clothing.

Nancy selects an outfit at random, by first choosing one blouse from Wardrobe \(A\), followed by choosing one pair of pants from Wardrobe \(B\).

\(X\) denotes the score of a randomly chosen outfit.

• If the blouse and pants have the same formality, then the score of the outfit is the sum of their respective indices. For example, the score of an outfit comprising \(T_4\) and \(P_3\) is \(7\).
• If the blouse and pants have different formality, then the score of the outfit is the product of their respective indices. For example, the score of an outfit comprising \(T_2\) and \(P_3\) is \(6\).

1. Show that \(\mathrm{P}(X=4) = \frac{1}{6}\).[1]
2. Find the probability distribution of \(X\).[3]
3. Show that \(\mathrm{E}(X) = \frac{55}{12}\) and find \(\mathrm{Var}(X)\).[3]
4. Nancy selects three outfits in succession without replacement. Find the probability that there are exactly two outfits that comprise both a formal blouse and a pair of formal pants.[3]