2022 EJC J2 MYE P2 Q6

The diagram above shows a spinner with \(3\) equal sections, labelled “\(1\)”, “\(2\)” and “\(3\)”, and an arrow which has an equal chance of coming to rest over any of the \(3\) sections. The score obtained from a spin is the number of the section over which the arrow comes to rest. In addition, a bag contains \(2\) red balls and \(3\) black balls. In a game, Roy first chooses a ball at random from the bag and then completes two spins of the spinner. Let \(X\) be the random variable denoted as follows:

• If Roy chooses a red ball, \(X\) is the absolute difference of the two scores from the two spins.

• If Roy chooses a black ball, \(X\) is the smaller of the two scores if they are different, and the common score if they are the same.

1. Show that \(\mathrm{P}\left( X=2 \right)=\frac{13}{45}\).[2]
2. Tabulate the probability distribution of \(X\).[4]
3. Find \(\mathrm{E}\left( X \right)\) and \(\mathrm{Var}\left( X \right)\).[2]