2018 TPJC JC2 MYE P2 Q6

Four objects \(a\), \(b\), \(c\), \(d\) are to be placed into four containers \(A\), \(B\), \(C\) and \(D\), with one in each container. An object is said to be correctly placed if it is put into the container of the same letter (for example: \(a\) into \(A\)), but incorrect placings otherwise. Let \(X\) be the number of correct placings and \(Y\) be the number of incorrect placings when the four objects are placed at random into the four containers.

1. Show that \(\mathrm{P}\left( X=2 \right)=\frac{1}{4}\), and find the rest of the probability distribution of \(X\).[5]
   
2. Find \(\mathrm{E}\left( X \right)\) and show that \(\mathrm{Var}\left( X \right)=1\).[3]
3. Deduce from the results of part (ii) the values of \(\mathrm{E}\left( Y \right)\) and \(\mathrm{Var}\left( Y \right)\).[2]