N1990 P2 Q7

IB AA SL: core and optional content

Core: event probabilities, distribution and expectation. Optional: variance.

A circular card is divided into \(3\) sectors scoring \(1, 2, 3\) and having angles \(135{}^\circ \), \(90{}^\circ \), \(135{}^\circ \) respectively. On a second circular card, sectors scoring 1, 2, 3, have angles \(180{}^\circ \), \(90{}^\circ \), \(90{}^\circ \) respectively (see diagram). Each card has a pointer pivoted at its centre. After being set in motion, the pointers come to rest independently at random positions. Find the probability that

1. the score on each card is \(1\).

2. the score on at least one of the cards is \(3\).

The random variable \(X\) is the larger of the two scores if they are different, and their common value if they are the same. Show that \(\mathrm{P}\left( X=2 \right)=\frac{9}{32}\) . Show that \(\mathrm{E}\left( X \right)=\frac{75}{32}\) and find \(\mathrm{Var}\left( X \right)\).